Actually, that is precisely why they didn't take off: Without push notifications a lot of mobile applications are simply not feasible. If something is not feasible, most people will have no reason to invest in ux. It's self-fulfilling.
However, with push api, cache api, being installable (and maybe some additional apis that elude me right now), I don't see any specific thing that would not allow carefully crated web apps to have great ux on mobile.