Every decent host OS already has a dedicated driver stack to provide game controller input to applications in a useful manner. Why the heck would you ship a reimplementation of that in JS in a website?
If it hasn't been invented yet, you don't need driver software for it, do you? ;)
Anyway, in your scenario the controller would be essentially a one off and you'd be better off writing a native app to interface with it for the one computer this experiment will run on.
Unlikely. The convenience incentives are far too high to leave features on the table.
Not unlike the programming language or the app (growing until it half-implements LISP or half-implements an email client), the browser will grow until it half-implements an operating system.