We should probably stop calling them "web browsers" at this point. We've basically re-implemented the idea of an Operating System in the browser, so we now have a full-fledged OS like Linux, BSD, Plan9, OS/2, Minix or whatever, being used to host a "poor man's OS" which is recreating most of what the base OS does!
To say "things have gone very wrong" is a dramatic understatement in many ways. I still think we messed up by migrating away from the mobile-code approach for apps. But I largely blame Sun for that, as they screwed the pooch by shipping the Consumer JRE about 7 years too late, ignoring needed features for doing desktop and interactive apps (like modern audio and video codecs, etc.) and didn't stay on top of JRE security. Now, Java, which was probably the best "mobile code" platform that ever got mainstream traction, is basically dead as a client platform. :-(
Browsers are great for doing what they were designed to do: Browsing hypermedia. But to shoehorn "remote application delivery" into the browser seems like a step sideways (at best) to me. We're layering hacks on top of hacks on top of hacks now, to try and recreate the "app" experience inside the browser. This strikes me as sub-optimal.