EDIT: You know, after some time to cool off, Google Authenticator 2FA can still be enabled and isn’t being phased out like the less secure SMS 2FA, so it’s really not the end of the world here. The chance of permanent lockout is avoided, even if the whole Google Prompt system is still wack.
Using your device to do whatever is op’s right. From reading the post, it seems to me that the problem is that they disable other forms of auth. This is for sure intentional, or at least a low priority bug for obvious reasons. I had the same issue, but it was failing to pull up the menu in my stock nothing phone 1. It got fixed later, but why are my backup emails or phone numbers not being used as other forms of 2fa. That is when I realised that despite my efforts, I have ended up relying on Google too much. In the process of changing that, even if it costs me money to host the servers.