Safari (and Firefox) does not have https://developer.mozilla.org/en-US/docs/Web/API/BeforeInsta... So you can’t have a custom button/way to prompt players to install.
However, they can install a PWA manually, by themselves; there is the option somewhere in their browser menu.
If Apple supported this API it would be much easier to convince stakeholders that PWA is enough.
You can do browser detection and show a dismissible pop up with instructions but it will still just confuse some users and others with scoff at the need to do that.
“it’s just opening a browser window with no actual browser controls”: yes, due to the `display` setting in the app manifest: https://developer.mozilla.org/en-US/docs/Web/Manifest/displa...
It’s similar to a `WebView` component: https://en.wikipedia.org/wiki/WebView
But there is nothing a PWA can do that a website can’t do. It’s all about Web APIs. Take push notifications (https://developer.mozilla.org/en-US/docs/Web/Progressive_web...): they make more sense for an (PW)app, but a simple website can do them too (if given permission).
See MDN intro on PWA: https://developer.mozilla.org/en-US/docs/Web/Progressive_web..., or https://web.dev/learn/pwa/ for more info.