There is no intrinsic reason for the browser to be so central, but there is a definite historical reason. In the end it is simply how things turned out.
At this point, Mozilla has invested a lot of effort into Firefox. Version 3.0 of the browser will have DOM storage and offline execution. This is a highly advanced FOSS project, with a lot of momentum. To match their significant technological achievement and market penetration would be a formidable task indeed.
The above reasons is why I suggest adding X Window manager features to Firefox, making it the central mashup point for both SaaS and local applications.
There is definitely a valid argument against putting too much in the browser, but what is the alternative? Maintain two separate desktop environments: the browser for SaaS, and a traditional X setup for local applications?
Note that currently, the X-based desktop is the unifying construct, and the browser just another application that runs on top of it. Firefox, however is not just an application -- it is on the verge of becoming a platform. So, I am proposing to better integrate these two key systems, by making Firefox act like an X window manager.
Of course, as you earlier pointed out, we do need to worry about many things, especially state management and the network protocol. HTML/HTTP is not the ideal solution. It should be possible, however, to solve these problems at higher protocol layers. This approach may not have the best architecture, but it avoids the monumental task of recreating Firefox's functionality elsewhere.