Comment by greatgib
It is not that you have to "interact" with Google the problem, in the sense of interacting like downloading an app. You can use the Aurora store, but once you try to use the app, the app itself will redirect you to an oauth2 login for your Google account, the kind that is associating your "phone"/Google service globally with your Google account. And this despite the fact that I will only use password login for openai and mistral, that should not be linked to Google anyway.
In addition with integrity verification, I can easily think that they are using it for "push notifications" that will also travel through Google.
So, it is not only that you will have to "interact" with Google, but the fact that you will be forced to let Google track you: which phone you use, which ip, which app with which account, used when, where, ...".
That defects a little bit the purpose to have a "free" phone if you still have to give your data to Google.
So the problem is the "push" not the "pull".
Aurora Store also runs a bunch of their own throwaway Google accounts you can use (the “anonymous” option on the sign-in screen). Usually works great, though sometimes takes a few tries to get a working account.
Many apps do require passing the integrity check, though, but microG is getting better on that front (and IIRC you don’t need a Google account for that).