Earlier quoted context omitted.
I think a more radical departure is needed. It's about time to acknowledge that the web is increasingly being used to access full-blown applications more often than "webpages." Web browsers have more or less become mini operating systems, running elaborate virtual machines. There's way too much complexity for everyone involved — from web devs and browser devs to the users and the people who maintain the standards, th…
In my opinion, if there is something that the web can't do as well as native it is a bug. Web technologies can already do most of what you are proposing, including notifications. There are some performance issues, but they are well on their way to being fixed.
What we would need, if the browser is to become a platform for actual productivity tools and not shiny toys, is a decent persistent storage interface - one that would be controlled by users, not by applications, and that could be browsed, monitored. And most importantly, one that would be reliable. And then, on top of that, a stronger split between what's on-line and what's off-line. Because some data and some tasks should really not be done through a network.