This made me realize how little innovation there is in the browser space today.
I don't think that's true. There's lots of innovation, but it's in things that aren't necessarily obvious to the end user until there's decent levels of adoption by developers. Websites can access gamepads, advanced hardware-accelerated 3D, bluetooth, accelerometers, different layout engines, P2P data sharing, push notifications ... the UI that doesn't change very often, but that's only a tiny aspect of what a browser is.