Since it won't happen the other way around, how do I get people to approach ME in public?

Wait, who said gender roles are dead? That's not a thing. Assuming youre a man, most women that are interested in you (or might be interested in you in the future) will expect you to make the first move. I don't think that's the patriarchy at work; most straight women want a man who is confident and assertive and I don't think there's anything problematic about that.

Literally wear a name tag.

A guy, that goes by the name of etymology nerd, did this everyday for a month and people approached him more often.

He made a short video explaining why this was the case and how it was an invitation to converse.

You seem to be confusing what you want with gender roles. Nobody said that men must make the first move. Many satisfying relationships start because women make the first move. But by consequence of the fact that you want a relationship, it naturally then follows that you'd have to make the first move.

IMO using props is a poor move. Might get your foot in the door, but it'll be obvious that your interest/commitment to the prop is not genuine. People can tell if you're acting, so I would ignore any comment that tells you to imagine and act out a scenario - doubly so if you're using a prop.

The trick is to realize that a cold call almost never works. There is a very low chance that any one person you run into on the street is looking for a relationship, and an even lower chance that they'd be willing to bet on a stranger for that relationship. So you're facing 2 filters that are lowering your chances that any one person you meet would want to get into a relationship with you.

You can't affect the first filter, but you can at least change the second filter - just don't be a stranger. It's easier said than done, but it's possible with concerted effort. To put it bluntly: be amicable and be social. Put yourself in situations where you meet people, and befriend them. And you are by far more likely to run into a potential partner from the people you already know than in a public park. I don't even mean to pretend to be friends - I mean actually be friends. Socializing has a compounding effect where the more you socialize, the more people you get introduced to. That's also important because of the fact that you can't affect the first filter. Clearing the first filter is really a numbers game, to simply know a lot of people.

You'll need to learn to maintain a social circle. Based on what I can tell, you seem to either be an introvert or have social anxiety. And honestly, I understand. I can't say that the process will be easy or that it's fair that extroverts have a leg up in the process, but the unfortunate reality is that society is built upon the assumption that people are social, and you have to play by those rules. The upshot is that more people are willing to be in a relationship than you'd think, and you don't really have to expand your social circle that much or maintain it that well before you come across someone who agrees to be in a relationship with you.

Since chatting someone up is out of the question for me

Why? Are you unable to speak? Are you unwilling to try? Because it sounds like you're hopeless about the idea of meeting someone and you don't feel like your own efforts could possibly help.

I don’t see why the man must start this dance.

The truth is that nobody, regardless of gender, wants to put themselves in a position where they could be rejected. And women usually don't need to. While you're leaning against the wall waiting for that woman to start talking with you another guy will approach her and ask her out. We all want to feel like someone is interested in us. It's flattering. It makes us feel good. And your question makes it clear you're unwilling to put the effort into showing you are interested. You want someone else to show they are interested in you. That way you don't have to risk anything. But low risk means low reward.

This may sound harsh, but I do understand where you're coming from. I went through this myself. One thing that helped me was learning to let go of any expectations when approaching and talking with someone. Try to understand you aren't there to pick her up, seduce her, or whatever. You are just there to talk. Everyone is interesting in some way. Practice asking people about themselves. You'll be awkward at first. You'll mess up and be embarrassed. But keep trying. If you can let go of your own desires and just be curious it will take you far. And talk with everyone you meet. Men, women, old, young, don't discriminate. It'll help you realize it's just a conversation.

Also, put one of those sticky name tags on your shirt. "Hi my name is Tudsamfa". It makes people more comfortable knowing your name.

Have you tried falling off a ladder. Anyone who doesnt run to offer assistance is not someone you wanna prone bone anyway. Time your ladder fall as the hottie approaches.

chatting someone up is out of the question for me

Some basic life advice for you: focus on things you can control, not on things you can't. Approach the people you want to interact with. That's completely under your control. You can't really control whether other people approach you. Spending your effort on that is a waste.

Maybe go on a site like meetup or find local events you are interested in. Go out and do something with strangers: help a charity, go on an organized hike, take a language class, do some volunteering.

Just hanging at the park putting out the vibe likely won't get you where you want to be.

Go to the singles park on a bicycle. Ride somewhere with cute single girls in your area. Fall on the bicycle and hurt your knee. Wait for help. If the cute girls don't help, they're probably not that nice. Phew, you just saved a bullet there.

Is it this kind of stupid methods you want?

Okay, buy a shitload of oranges. Put them in a bag. Cut a hole in the bag, so it barely keeps together. Go to the park and find the cute girls again. Give the bag a firm shake so it falls apart and the oranges start rolling all over the place. Look sad. Attempt to pick up the oranges but keep dropping some, until they come and help.

Now for the next one, you will need to get out of your comfort zone. Go to the grocery store. Find the laundry detergent isle. Keep staring at the products until a cute single girl comes by. Look confused. Say to her: "excuse me, I'm really confused about all these, which one are you getting?" Eventually thank her for help and turn the conversation into something else " I'm new in this town, do you know any other great places to get laundry detergents (or whatever)"

Oh, I forgot to mention, you need to be wearing a suit or at least a blazer and proper shoes. None of the above will work very well if you look like a hobo who washes with a toilet brush.

What you need, my friend, is a slump buster. Casual sex with a woman you otherwise wouldn’t be interested in. I don’t know if it’s a sixth (or 7th or 8th) sense that women have, but they can tell if you’ve gotten any recently. If you have, it makes you more desirable. Alternatively, a good wank before going out in public, but that’s not nearly as effective. I ended many droughts with a slump buster back in my single days.

If you're a dude and you wanna be approached by regular women and not gold-diggers, you kinda have to be cute. If you're not too handsome you will have to go outside and wait a lot, I guess, even more so if you don't give off "approachable vibes". Or you could do what most men have done since time immemorial: learn a bit about people and what makes them smile and go do that regardless of your fears.

And what is this nonsense about being progressive and "gender roles". Women are scared kittens and the mere possibility of rejection, public or not, wrecks most of their psyches. And men are hornier (else the prostitution market would look very different, lol), so they know they'll be approached eventually, at least in their younger years. You're not battling ideology, you're battling economy! You will have to make a move, unless you're handsome enough that a woman makes a cost-benefit analysis and finds the risk of EMOTIONAL DAMAGE worth the prize. You might be! If so, well, go outside. If not, you're gonna have to be brave and do the unthinkable: talk to women and risk ridicule. Most women are nice, sweet and non-confrontational, at least IME, and if you know the least about women you can tell the bad apples apart easily, so give it a go.

Wear a button that says "I'm approachable!"

By being interesting.

How you do that is up to you. Like visibly enjoying hobbies, having an eccentric look, going for a fun personality, or being in cool locations. It also allows you to talk about something you personally enjoy.

People here talking about "props", and that's fair and dandy but only and only if you truly like or are into said props. Because you will attract people who are into that too, but also repel those who aren't. As a woman I never cared much about either people with dogs or babies. None of my business. But I have approached people with other animals or doing arts for example.

My recommendation would be to join a group activity and stick to it for a couple months. See how it goes, then switch to something else. It can be life drawing, bushwalking, patisserie, guitar, board games- you name it. This puts you back in control because the activity acts as the prop, and by context you are allowed to say "oh such nice drawings, how do you do xyz?" " Are those shoes good for walking?" "Do you bake often?" You start the conversation and it doesn't need to be personal at all. I'm always surprised to see normal people cross the line and start asking personal questions about me once I make it obvious that I'm open to conversation.

Until very recently, I had neon colored hair. Pink, blue, green, orange, purple, red; the whole rainbow. Changed it once every few months.

Women very frequently came up to me to compliment my hair and often would lead to conversations about where I got my dye, who did it, etc. like women stopping their cars in parking lots just to give me a compliment. More often than not it turned into a short convo, that usually ended around the time I mentioned my wife was the artist who did my hair.

Had I known this was a thing before I was married, I would have died my hair in a heartbeat.

I think it may have been a little disarming and was something unique enough that it was worth striking up a convo.

Wear a t-shirt that says I'm single you can talk to me.

I'm kidding. Sort of.

Use “social props.” I’m not sure if that’s a real term, but like the colored-hair guy said, it’s about creating an easy conversation starter and giving people “permission” to approach you. An interesting hat, reading a book where people can see the cover / title, a pet or a friend’s pet, doing a hobby like painting (you don’t have to be good at it), comic books, musical instrument, D&D manual, playing dominos…whatever! If someone wants to chat, just give them a small natural conversation starter and frequently look up from it, take breaks, smile; so they know it’s ok to interrupt you. If someone shows interest by a prolonged or repeated glance, just smile and say, “Hey are you into ‘whatever’ too?”

girls will signal to me that they might be open to having a conversation, but it's rare that one would directly initiate. the trick for me is recognizing the signals when i see them, and not hanging around for too long especially if it doesn't look like it's working out.

Try to look like you might have cocaine on you.

(Sorry man, I got nothing 😔 )

I’ve been approached by a woman exactly one time. We’ve now been married for 21 years.

Get a cute dog. Or a cat on a leash. Give them something to talk about.

have you considered purchasing (or renting?) a Horga'hn?

Probably the best thing I ever did to get random people to talk to me was growing a big curly handlebar moustache, now complimented with a long bushy beard.

My fashion choices also tend to make me stand out a bit- brightly colored Hawaiian shirts in the warmer months (I have one with pictures of the dog breed I have on it, that gets a lot of people approaching me,) occasionally a kilt (people love to ask about the kilt) interesting sunglasses, hats (used to wear a bowler occasionally, I'm less of a fan of it these days, panama hat in the summer, etc.)

Clothing and style choices are a little tricky. There's kind of a fine line between wearing something interesting that makes people want to talk to you and coming across as a fedora-wearing neckbeard who's trying too hard. Those choices have to look good on you, you have to like them and give off a bit of confidence while wearing them, and it has to be something that will catch the attention of the kind of person you want to attract.

And most importantly, you need to be able to carry a conversation from there. That's the hard part.

Having some story or a joke at the ready is a pretty good crutch to kind of get yourself over that last part. For example my go-to when people come up to me to compliment my beard/moustache is to joke that "I grew it myself" which is usually good for a chuckle, and then the ice is broken, and you can kind of try to steer the conversation from there.

I've had a lot of fun conversations with strangers and made a few friends along the way. I never personally had much luck turning that into a romantic relationship, but that was also never something I actively pursued much in general, I just kind of let things go from there and through friends who I met that way I eventually met my wife.

be in the moment

speak about the moment

if that moment isn't happening, dont force it

You have to talk to them.

Try to just be friends. Try to be funny. Try to be a little irreverent. Unpredictable. Amusing.

Nice day huh?

Did you see that cool show / game last night / dog that ran by just now with a six pack of beer?!

Excuse me you look really familiar. Have we met before?

Hi I’m Fred what’s your name? (Only do this if your name is Fred otherwise it gets awkward)

If they blow you off whatever. That one was just practice.

You got this bud!

Step #1 - Be attractive.

Step #2 - Be very outgoing.

Step #3 - Be ridiculously charming.

Do this, and you will attract interest from random strangers who see you being wonderful, and want to be involved in whatever you are doing and / or saying.

Do something poorly/dangerous in the gym, attach yourself (politely) to the person who comes to help,

Or do something you're passionate about and offer your knowledge to people who are earnestly trying their best but not doing it well (be conscience and ask if they'd like your experience to guide them)

Alternatively go hang out at an old folks home, they'd love your company, they'll regail you with stories from their glory days, and they'll probably talk you up to their family.

Volunteer your time at animal shelters/public service events.

Get a dog, or a child; they’re chick-magnets. Even if not your own. Maybe you could take a nephew/niece/whatever to the playground once a week? Walk the neighbour’s dog for a few blocks? As someone else said, once there’s a prop, they’re not approaching you directly, they’re interacting with a common interest, and that’s a good way to meet people.