I am in the Netherlands and lately AC's are getting more common. As someone with an appartement top floor flat roof, living room faced south the split AC installed this year is a blessing. Summers get hot nowadays.
Our guy said splits are great for 2 or 3 units, any more a heat pump is more cost effective so that's what we went for. Went with a city recommend installer and they were awesome.
Split systems use heat pumps. I think you might be talking about a wall- or ceiling-mounted split system vs a ducted system. It's the same technology for both.
I'm sure it varies a bit by region, but that wouldn't be a crazy price for the NE. HVAC is just expensive these days.
I'd recommend getting at least three quotes. If different companies propose different setups (e.g., all heads vs ducted vs mixed), try to get at least two quotes for each. I ended up getting six quotes due to how much they varied in both price and design. My highest and lowest quotes were more than 20k apart for a <1500 sqft space.
If you have access to a 0% financing program, don't wait too long. Those run out quick because it's such an amazing deal.
I had a similar situation (except baseboard heat instead) and ended up with a mixed system. It's really dependent on layout. I'd be careful with anyone who doesn't take a good look around before giving you a recommendation, unless you have zero room for ducting for instance.
Most ACs are reverse cycle these days since it's a very minimal extra cost to allow it to both heat and cool - it just needs a four-way reversing valve. There's no point making it only cool when you can instead make it both heat and cool for a similar price.
The USA is weird though. Companies still make units that only cool, and strangely there's a big price difference between cooling-only systems vs reverse cycle systems. I haven't seen this in other countries.
Because heat pumps are popular now, and the government is helping people pay for them through inflation reduction act rebates. So they can jack up the price and price gouge the shit out of us, because they know they can and know that they will get away with it. And I bet they were never popular because the same people selling home AC units also sell gas powered furnaces for home heating. So they charge you for two appliances and two installations, with two maintenance calls every time something goes wrong and two upgrade cycles. But now that they come as one unit, they still have to find some way to return more money to shareholders than last quarter so the price goes up.
I just got whole home AC replaced with a heat pump with an integrated furnace backup because it can get very cold here, and also had all my baseboard heaters ripped out. Perfectly fine in the winter and it has freed up at least 1/4 of all my walls to have stuff right up against them if I so choose, but really annoyed that the whole thing costed $20k when it could and should have been built this way 60 years ago. Not to mention all the inefficiency of burning gas for heat when heat pumps move more energy than they consume, multiplied across decades for nearly every building on the planet.