You can still run an Android build that doesn't require a Google signature for apps. You'll just lose access to Play Integrity APIs, which you wouldn't get from non-Android Linux phones either. A better technical solution is to set up a federated replacement for Play Integrity that third party ROM developers can opt into and a library that can use that or Play Integrity for app developers that want it to use.
Banking apps will not work then.
That's a bit overblown. Almost all banking apps work fine. You might be one of the unlucky few of course, but there's no need to scare others from running free software.
I think the "one smartphone for absolutely everything" era is over. Either switch banks (there are many who don't do this nonsense) or have a dedicated Android/iOS device for banking.