Everything is moving to the browser but the browser now requires a lot of capabilities from the hardware and drivers: 3D acceleration, sound, printing, power management, etc. Effectively all the problem areas for Linux outside of custom hardware (like phones).
I've never had problems with any of those on Linux. YMMV, of course, and I may just have been lucky.
While you might have problems if you build a custom machine or use extremely recent hardware (which doesn't have good drivers), the general implication of browser-as-OS is that people wouldn't be installing this new system on old computers, but instead would buy new computers with it already baked in. Manufacturers, if they wanted to succeed, would be forced to make sure that all the drivers and the system in general worked well.
Which covers most/all of those things. Reasonably new and innovative hardware (like GPU switching in laptops) just doesn't work, period. There are plenty more examples. The browser makes Linux worse since the only thing you need from your underlying platform is really good driver and power management support.
The only issue on the driver list I've ever encountered is wireless failing after suspending. Oddly, that issue only showed up after my latest kernel upgrade (and is a reason that I'm not using my netbook much until I figure out a fix). I suppose that I've just been very lucky with the hardware I've installed Linux on.
(I've also removed the FUD comment from my earlier post).
