Possible yes, probable not so much. Building extensions is simpler than building desktop apps. For one you need only JavaScript. While if you move in the OS you'd have to support different versions of systems (you have Windows, OSX, Android, IOS, Linux) and it is much more cumbersome than in the browser.
Interesting concept. Asking as a layman: is this possible? Is the browser sandboxed enough away from the OS that this is a difficult problem? Is that why we haven’t seen solutions that implement this approach?