Touch-based (and by extension, NUI-based) OS'es and mobile applications are the future. Windows always sucked. Mac OS always sucked. Every desktop OS ever built sucked because it is a horrible way to use a computer. Nobody ever really wanted to use these terrible desktop metaphor systems... they only ever did because they had to.
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.
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.