Chrome has that sorted already on android, where you can run up a site with a home screen link, different icon and separate listing in the task list. Given the work they've recently put in on full screen mode on desktop it looks like they're maybe a couple of versions away from having something similar on desktop
They're actively adding features. It's a little more complex than it used to be with the manifest, and needing a service worker and encryption for a lot of features, but it's all still there
