The real question is what you’re trying to accomplish and why.

That informs your choices.

Privacy, security and anonymity are different things.

Is this more of an argument of privacy vs anonymity?

Exactly!
Privacy and anonymity are two seperate things. You can be post things publicly and be anonymous. You can post things privately to people who know you.

Having your credit card would prevent anonymity, not privacy.

There are services that allow anonymous payments. I think Proton, Mailbox.org, and Posteo all accept cash payments.

Kagi has a "privacy pass" feature, which uses some fancy ephemeral authentication tokens that are not traceable to your specific account. You lose personalized settings, naturally (like site rankings) but you can do searches that aren't directly tied to your account.

For email, there's no one-size-fits-all risk model. I mean, my email address is my full name, @ a domain that I own and is itself traceable to me. I have no need for anonymity, but I DO need privacy and security.

you can pay for proton via cash in mailbox (sending cash to their address with account details)

bitcoin isn't cheap

Well, you dont have to buy whole digit BTC (or any other cryptocurrency), you can convert like €50 into the wild and sketchy world of unstable currency which will be around 0.0026436362 BTC, but is enough to cover the cost of the transaction. I'm not sure I recommend it, only that it's technically not expensive because you're still using the equivalent of your currency, with a transitional currency.

things like search that limits options

And you can self host SearXNG, metasearch engines are much easier to handle than search engines, and I believe they are private enough -- no tracking or sign in data is communicated through each search API -- but I do not know all there is to know.

I guess when you pay with a credit card, theres limited information being shared. Taking things at face value, you are paying for a service that wont track you, see your data, give away your data etc.

Looking at it in terms of privacy vs. anon is a good way to look at it. If you want anonymity, you won't be using a credit card I assume, unless it's a throwaway you can top up. I imagine people who want full anonymity would be using stuff that's more specialised and niche than the likes of Proton etc., likely FOSS stuff, home servers, services without login/registration.

Privacy is like a step away from anonymity in that regard I think.

I think the primary use of a paid Proton or similar situation is to deprive the advertising panopticon of that data, not the state beyond the point of even an executive warrant, much less judicial. Your confirmation email from your dentist inherently deanonymizes the recipient as you, so you wouldn't want that to go somewhere anonymous, but you also probably don't want to feed the ad data beast by sending that to Google. It's no protection against the state, but the state can probably lean on your dentist if they care too much about when that appointment is, but using a Proton Mail combined with their aliases can deprive the private data brokers (who are the main source for low level ambient government surveillance) of a lot of information about you.

Here are two ways to pay for things without the payment method being a direct line to you:

Use cash. People will say stupid things like “I’m European and live in a surveillance state and they track the serial numbers!” Let me introduce you to something called “money laundering”. What you do is ask for change or for a big bill. You may need to be a normal person, like a customer of some sort, or a non schizophrenic shut in computer nerd if asking of some person (like a bartender getting off at shift change!).

Use a gift card. People will say stupid things like “well, they know you bought a gift card, so they can tie it back to you!” Again, allow me to introduce you to the vagaries of something called money laundering. Have someone else buy and activate the gift card. Heck, if you really want to be a weirdo about it, do it with your friends and put all your other person purchased gift cards in a fishbowl and draw random ones out.

Gift card stuff is something to poke and prod at first because different stores and cards have different ways of adhering to kyc.

You're still using someone else's computer.

Anything you pay for will only be as anonymous as the payment method you use, things like bitcoin are not really anonymous since buying bitcoin without know your customer regulations is hard. If your adversary has the resources or your threat model is high enough even paying by mailing cash proposes a risk which is why things like tor need to be free. My advice is get a password manager and create unique long passwords and never reuse usernames to prevent doxxing threats, I use keepassXc for everything even my bank.