That's an interesting idea that I think I agree with, or maybe the site can only prompt you if it's been added but you can still manually enable it if you want (though adding the homescreen isn't the end of the world if you want notifications). That said, PWAs have downsides on iOS and I can't speak for Android. I'd almost always prefer to stay in Safari when looking at web content. I'm ok with webviews in apps for their internal UI but I almost always bounce from the in-app-browser to safari with the exception of maybe something like Reddit (Apollo) since normally things I look at from there are news articles or other read-only-type content. As soon as I need to login then I'm off to Safari.

Off topic: I don't know if the situation is better on Android but my biggest pet peeve currently is being bounced into an app instead of the web version of the same page. Like clicking on an IMDB link in google results or a Steam link (often comments, like when I'm trying to figure out if something is a bug while playing on a console). I know the workaround for this (long-press link, open in new tab, or delete the app) but on the other side of that coin sometimes I get sent to the Amazon webpage but I want to bounce to the app where I'm logged in and the UI is better (IMHO) but I've yet to find a good way to do that (maybe "share" the page to the Amazon app? I've never tried).