From ente Authenticator launch blog post, they do state it is open-source.

My recommendation is to stay away from the auth services that are account based such as ente or bitwarden UNLESS you have a 2nd method of authentication such as a yubikey. Why? Because you obviously can’t store, for example, your bitwarden 2FA code in your bitwarden vault. Same with ente. That’s why on android I use Authenticator Pro, as it isn’t account based and has great backup options, such as backing up to cloud and file encryption. Not sure what an equivalent would be on iOS but I’d definitely want to find out! Bonus points if it’s available on both iOS and Android.

You’re correct but you can still use it without ever attaching an account to it, which is preferrable.

Don’t ditch your pixel, as you can replace the Stock Android on your phone with GrapheneOS instead. It’s still android but more secure. Google Auth is probably one of the easiest to ditch. For something like Authy or Authenticator Pro, although you won’t have to do that. Google’s authenticator is good enough really, even from the privacy side of things. Maps and YouTube though? Those are indeed tough to leave behind.