Buying a flight or ordering food via a PWA would just be a web site. PWA doesn't necessarily imply that you "install" by adding it to your home page. It could just be a web site where you order something and permit push notifications.

That is fine, as long as you offer an alternative to push notifications. I have them permanently disabled in my browser so that each website I visit does not keep asking if I want them to spam me.

Notifications from browser... No thanks

That's what PWAs are is a website that allows users to install a WebView icon on their desktop if they choose to. I think this is the advantage of PWAs, they are merely websites with a manifest and install option for users that may want an offline experience, native notifications, or some other feature that would not generally be available within a browser-exclusive context.

"install" is kind of a misnomer, it's more like "put a shortcut on the desktop" which has been around forever, even for links to websites with their own icon

Once you've granted a website enough permissions to run in the background and notify you when a food delivery has arrived, it's safely crossed into the "app" sandbox.

The OPs point is we're now blinded to all these website popups asking for permission, that the web app potential can't truly breakthrough.

As a corollary to the site permissions making it effectively an app, I like web sites in a regular browser because it enforces more limitations than what an app may try to get away with.