1) A truly user focused browser is created, the fabled "user agent". The ad-focused web doesn't support that browser, but websites that care about users do support it. Thus, people who want more than ad-drivel use the niche browser and have access to a web full of weird and non-profit-focused content.
2) Possibly a fork of the underlying technologies. Maybe the browser mentioned uses incompatible technologies or protocols. Maybe this new web is based on something other than HTML and JavaScript.
Probably not. It's a wild idea. It's probably too hard to do better than the existing technologies, and the effort required for such a fork seems ever less likely in this time of dissipating focus and hobbies.