Native apps will likely always have a place when OS/UI integration are paramount. But web apps will become a lot more prevalent than they have been previous as more and more hooks are exposed to the native core. Which is a good thing for everyone.
I don't really know the best way to delineate what should and shouldn't be a web app, beyond a vague notion that my daily news rotation should be in the browser.
Actually I'd argue the web is great for that too. On Android you can already implement a messaging app, complete with notifications, on the web.
I wish people like my bank used it. I don't want to install their native app because I barely ever use it, but would I like to get urgent notifications from them every now and then? You bet I would.
Just because there's been a lot of progress in making web apps do things that were more natural for native apps, doesn't mean that it's the best way to go about something.
I get the wish for more support for web apps, though. I just switched over from a near 4-year-old phone that was replete with security updates and other un-removable stuff in the storage which prevented me from having a lot of apps installed (especially as so many insist on a large footprint in internal storage), so I increasingly used my web browser to access things. After the switch, though, you can bet I downloaded a bunch of apps I had deleted.