And Google has an answer to the "just install the APK from somewhere else" workaround, too. Many apps now integrate a check that prevents them from running if they're not properly linked to the Play Store.
I had an app that I needed to use, and the only available log-in method was via firebase's SMS. Firebase flat out refused to allow me to login because of Google Play Integrity, and there was no web only option.
I ended up having to use my spouse's iPhone...
>Firebase flat out refused to allow me to login because of Google Play Integrity
Sounds like the issue is that you don't have play services installed, rather than play integrity specifically.