Websites can get Home Screen shortcuts; this has been a thing since the original iPhone.
iOS Add to Home Screen is a neglected bookmarking feature, buried within a context menu few users scroll into. It's effective, but presenting it as a viable alternative to Progressive Web Apps is unrealistic.
In 2020, lots of universities required students to install fairly invasive stuff on their computers for remote learning, but -- thanks in large part to Apple's policies -- their phones remained pretty much untouched.
Furthermore, Proctorio and plenty of other invasive testing software has no trouble running on a Mac. Your last sentance is about as coherent as thanking Google for stopping McGraw-Hill from spying on Android users.
Because the web is mostly useless if any of those are turned off now - even easy cases like static forms commonly break with javascript off rather than degrading to just server-side validation.
Thats not to say they shouldn't support additional web functionality for everyone, just that "Add to Home Screen" is likely never going to unlock additional native hardware capabilities or exclusive background processing modes.
This is also why every browser implements web extensions via a store model. The store model allows the browser to serve as a port for auditing and revocation on abuse, because that is what the capabilities of web extensions require.