There is clearly a use case for keeping the whole UI. But it is the major use case and should the whole product philosophy be based on it ?

I feel there will be more sites where the URL won't matter or where the user will prefer simplicity to control.

I use Google Maps 99.99% in PWA mode and never mourned the lack of the URl bar, especially as I can open the site in normal browser mode anytime I ever want the full controls.

What if you want to copy and share the page?

I'd be fine with a 2~3 click operation to get the URL.

Which is basically how it works in Chrome in PWA mode: a few basic actions (get the URL, print, cast, adjust zoom etc) in a menu, and an option to punt it to the full chrome if needed.

That’s a fine idea.

Every PWA starts in a webpage, and you have to manually install it as PWA. If it's important for you to see the url, use it in the browser and don't install it? Installation only makes sense if the website/app is build for it.

Nobody is forcing you to use an installed PWA. Use it in a browser.

Then use it as a web page?