As not a web dev, out of curiosity: What would be the drawback or problem with showing a header or pop-up on mobile browsers, offering to install them as (web)app? Then using the PWA functionality of the browser to do it.

I'm not a heavy user of those but the result of PWAs has always been an icon that's handled by the OS like if it was any other native app, and when opened it just behaves like the web browser in kiosk mode just for that website.