I guess the desktop environment was just a random choice of "this works, so just leave it in" from whatever distribution they started with.
Granted, browsers aren't without their security holes, but then again, neither are operating systems. Given the amount of effort being put into browsers to make them secure (especially Chrome), my money's on that.
This system goes at least 2 layers deeper. System itself makes sure that each window has its own desktop environment and can't see others. Hardware takes care about the separation between security containers the apps are running in. Protection of the app itself is just the first line of defence and is not going away, so whatever sandboxing exists in the browser still applies.
They are also talking about protecting hardware sharing from being used to cross boundaries which is another layer of paranoia (not unwarranted)
If you build a webmail client, you need to know all about these attack vectors, and you need to go out of your way to prevent your application from being susceptible to them. Websites are insecure by default.
I don't trust a web browser with my email at all. Not yet. If I were to use webmail, I'd make sure to set up a separate instance of Firefox to run it in, with it's own profile. I will continue to use Thunderbird for now though.
I'm not against the idea of using webmail, I just don't think the web is secure enough yet.