Add to Home Screen does give you fullscreen. But if your state is in the URL, here is something I found out the hard way: on iOS, a PWA opened from the home screen has its own localStorage, separate from Safari. My own browser game keeps data in localStorage, and things a user saved in Safari were not there after adding it to the home screen. They had to do it again.

Keeping the state in the URL actually avoids this. There is no storage to be separated. The cost is what perilunar said, one shortcut per routine.