> A web app experience is at the mercy of the network I happen to be connected to.

That's one of many things that web apps have been fixing. It's entirely possible to write an offline-capable web app.

> In addition, as a user, the "build once, run anywhere" feature of web apps is a disadvantage since it often means that apps are written with the lowest common denominator in mind and don't quite blend in with the look and feel of either major mobile OS.

They do, however, look like web apps everywhere, and if you primarily use web apps, you'll feel right at home.