Other than avoiding those services as much as possible, I use a second phone number for "machine-communication"= whenever I'm not giving my phone# to a person.
I'm in the EU, I found a provider in my country that offered a prepaid sim card (pay-per-use) with no expiry date, but never use the credit on it because it's free to receive sms. I turn it on in my dual sim phone whenever I need it.
Did not see any requirement of the sort in the fine print, but even if there were, it's fine as long as you pick the right provider. If I had to make the occasional call it'd be still worth it. There are also providers that will keep a sim active indefinitely as long as you "purchase" one month (as little as 5€) every 1/2 years (most importantly, they do not charge you into negative credit). So basically free to operate as well.
Honestly I do it mostly to limit spam, if I did it only for privacy reasons I'd have more than two numbers but I fear one might start getting noticed by the autorities at that point :/ sms is inherently unsafe and not private.
Depending on your threat model, your best bet would probably be to purchase a burner phone at Walmart or something with cash. Then only use the phone for verification purposes.
I couldnt find any sim cards that didnt require a CC to activate. I brought one that came loaded with $40 and still asked for a CC to activate which sucked. My threat model is not existent I just believe think every service I sign up to needs to know who I am.
Not sure about SIM cards but you can still use cash to buy prepaid phones along with an amount of minutes/text to activate. Like the kind of prepaid phones that you you see the vendors sell x minutes/text/data. e.g. I've never been asked for a credit card to buy a Tracfone. Some places like Best Buy will ask you for other information to sell it but none of it needs to be real (ironically Best Buy asks for a phone number to buy a prepaid phone, LOL). You do need to set up a Tracfone account online to actually activate a phone + plan, not sure about the other prepaid vendors.
I've been using Jmp. You don't get a sim though (at least for calls and text, you can for data). It goes through Jabber. Their app can integrate with the dialer for calls. Never gave any personal info. Only paid with BTC / XMR.
SMS is not private. JMP is upfront about it: https://jmp.chat/privacy. Still didn't have to give any personal info so better than the alternative for a basic privacy enhancement threat model.