Foggy For the three apps I installed this week from Aurora: […] I just get a 'login to Play Store' page. […] I don't get any popups that they tried to access Play Integrity
This isn't caused by Play Integrity, it's caused by the apps detecting that their APK was installed by something that's not Google Play Store. It's because you installed them from Aurora Store.
You should be aware that Aurora is insecure, and (as far as I know) doesn't give a privacy benefit over using sandboxed Google Play with a dedicated Google account created just for sandboxed Google Play.
Foggy I get 'this app is not available for your device'.
This is caused by Google Play Store and/or Google Play services using Play Integrity.