I don't think of it as "needing WebGL", but as something finally utilizing it for this. Just about anything gets better if you can find a way to do it on the GPU, at least when considering hardware easily available for consumers. The sheer performance that enables is a quality of its own IMO.
> it is very hard to reproduce the native look and feel of text selection, editing and stuff, and even if you manage to get close to it, there will always be some "uncanny valley" effect...
People who want what they already have can just use what they already have.
And it's not just about matching it... you can scroll butter smooth at 240 fps. Or maybe you could make undo leave traces that fade out where something changed, making it more easy to follow when just keeping ctrl+y/z pressed for a bit -- or maybe that would make it harder, but the cool thing is that you easily try out all sorts of crazy things things when something is in shader form that would take a lot more effort to get working at 1/100th the speed any other way.
Browsers zoom in and out in 10% steps. That's incredibly crude and jarring. Like smooth scrolling, there could be smooth zooming. Or the other way, you only change zoom once the user has confirmed the new zoom level -- you can't do either in a browser. And that's such a tiny thing!
Of course how the GPU prefers things to be laid out kind of funnels what you'll end up experimenting with, but that's absolutely an avenue worth exploring, until you hit a wall, and then you climb that wall and keep going. Please :)