The thing is that the web doesn't need to catch up to native. The web just needs to satisfy the definition of "quality" held by the majority of users, where "quality" is defined as "fitness for use".
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.
