It's not only possible, but almost certain that the web will achieve acceptable performance for 99% of use cases in the next 1-2 years. After that, the web ends up with greater "fitness for use", since they can be ephemeral (instant, on-demand, no installation required), which native apps cannot do.
The only place where native is likely to hold out for the long term is 3D head mounted displays like the Oculus Rift.
It bears noting that running background tasks in iOS is extremely easy for developers.
[0] Android WebViews seem to improve steadily, while always lagging a bit behind the real browser. iOS WebViews were crappy for a long time, particularly because Apple kept Nitro for themselves while third-party hybrid apps used a way slower JS engine. But iOS 8 apparently made massive strides here, and WebViews are right on par with mobile Safari (Nitro, WebGL, etc). Or so they say; I haven't tried it out yet.