The money that Mozilla makes from Google comes primarily from setting Google as the default search engine in Firefox. You can see lots more details on this here (2016 numbers): https://www.mozilla.org/en-US/foundation/annualreport/2016/
I think stating that Google controls Firefox is vastly overstating the degree of influence.
Bing already pays for users, and once you give it time to build up a profile of you so it stops returning reptiles and vegetables when you're trying to search for Python libraries, it's equal in quality to Google. If having a high quality service literally give you things to use it doesn't break Google's stranglehold on search, what will?
I think the default for some time was DuckDuckGo, and that helped raise them out of complete obscurity, but it didn't reshape the internet landscape.
I think stating that Google controls Firefox is vastly overstating the degree of influence.