Nice component! I like how it’s implemented close to where the share buttons are. Agree that we should integrate this into core and the android banner.
I have an old unimplemented mockup from a while ago that might address some thoughts mentioned here:
The idea with the text here 
was to have some sort of clear “what’s in it for me” — and literally show the site logo in an app-like way to make it a little clearer what this is all about.
That was mocked up for Android, which makes things easier because you can include a button that directly adds the PWA to your device.
Unfortunately iOS still requires you to go through their share sheet to get to this action… but I think we can have a button that triggers the share options? (we already do this for clicking the share icon on posts in iOS…)