By the time you handle most of it, all you've done is make a wildly more complicated version of the DOM.
Probably I prefer "native" Web Apps over recompiled "native apps" but like it or not, this will come more and more as people ask for more performance and more variety of programming languages.
And like you said Qt already knows that (somewhat, there have been tech demos showing this but no real applications built on it), but it still means that YOUR APPLICATION must handle all of that, meaning at best you are downloading all the code for all of that at runtime, and at worst you just won't handle it because most people don't want to spend the time.
Flash was awful, an accessibility nightmare, a security nightmare, was/is not supported on many platforms/devices, and was horrible on battery life. I think we can absolutely do much much better than flash, but the answer is not to throw out the entire DOM and all of it's benefits and start over, the answer is to fix the problems that are easily fixable and settle on a good set of standards that we can have the option of ignoring for performance reasons in specific domains like games.