This exists, it's called Ungoogled Chromium. Removing whatever misfeatures Google adds to Chrome is fairly straightforward.
The real danger to having a single codebase for the whole web is spec validation. Most web standards rely on two independent implementations. For newer technologies we could technically count WebKit and Blink as separate, but Gecko was providing an entirely separate codebase that isn't a fork of anything.
Remember how WebSQL was basically put out to pasture because it was just SQLite, warts and all, shoved inside the web sandbox? That's the sort of problem we'd rather avoid. Single-implementation standards tend to pick up bugs and misbehaviors from their implementation, since everyone winds up depending on implementation bugs rather than getting them fixed to match spec.