No, because Mozilla don't own them, Google does.
The problem is not Google having a browser, open or closed. The problem is all the influence Google has elsewhere.
Do you mean rolling a new protocol in Chrome without warning that magically makes Google faster because they rolled it out on their end as well? This already happened with https://en.wikipedia.org/wiki/SPDY.
As long as what they do is technically good and not too heavy-handed, developers won't resist. In fact it is likely to be welcomed. But the effect will constantly leave Google a half-step ahead of everyone else on the upgrade train. After all it isn't hurting developers, it is just leaving Google with a slightly better competitive edge.
Where can users go if they still want YouTube to work well?
Perception is reality here.
Right now people get updates to Chrome and ChromeOS from Google. Microsoft is free to fork it. But if Microsoft goes one way and Google another, Google's version is what gets used. And everyone knows it. So anyone who wants to influence the direction of Chromium will talk to Google. Anyone who wants an idea of what is coming, will ask Google. Microsoft is a distinctly second class citizen.
Doubly so since Microsoft's last two attempts at a browser (IE and Edge) are now failed also-rans. If it attempts to fork Chromium, nobody is going to believe that their third try will win.
In its younger days, Microsoft fought hard to avoid being in such positions. For a flavor, read http://gunkies.org/wiki/Gordon_Letwin_OS/2_usenet_post to see how they thought about ownership of standards. But Microsoft has decided to cede this particular battle. At least for now.