Mozilla could already do a lot for privacy
For example they could prohibit cross domain javascript, so a website wouldn't load malevolent Javascript from other places by accident. They could evaluate every new W3C proposal and refuse everyone where the privacy implications are higher than any potential use (see DRM, WebASM, etc)
Instead Mozilla acts like any other company. Why? Because it has grown to a size where it's just like any other company. Their income comes from search engines as well as donations. Once that income breaks away, the whole project will fold.
What we'd need would be a radically more simple web replacement. Something where you can implement a fully functioning "browser" in less than a week, and new features are hard to add so it'll stay small. Perhaps something like VNC could be an answer.