Maybe someday, and it's definitely an option. Still, the Flash player isn't open source. And yes that matters. Also, Linux is still a second-hand citizen from Adobe's perspective, so far as I can tell. Much like Mac is still second-hand from Sun's perspective. Firefox is the only platform that stays on top of things, but their feature set and performance isn't quite there.
And there's the nice options for using Flash and Silverlight, for example, to implement HTML 5 features like the video tag.
But when it comes to apps (not sites), I still like Swing and Web Start personally, no matter what some people say. I just have to target years old versions due to that Mac issue.
In the future, things may change in who knows what ways, but that's for the future.