In Safari hit the share button then scroll down and hit Add to Home Screen.
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.
“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.
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.