I guess this is an example screenshot because of the sensitive content? do you too see the message arriving in the ui element?
I’m not sure if this is an SMS or arriving via firebase messaging. That would be different APIs to fetch the verification code.
If you run adb logcat while interacting you could see some class calls happening or failing and classify or attribute the problem for the microG issue tracker.
There are parts of the firebase auth (in firebaseui class calls) that are not implemented yet, but from the screenshots and without installing I can’t tell where it is failing.
Autofill is a android system service outside google play that can have multiple registered providers (like a password app) - so in general does work.