Nope. According to their financial statement, the Mozilla Foundation had $69M in "cash and cash equivalents" at the end of last year, about $329M in investments, and literally gave away millions of dollars in grant funding. It's not entirely clear to me how the financial interaction works between them, but the privately-held Mozilla Corporation (i.e. the 1000+(!) employee company that actually makes Firefox, and which the Foundation owns as a subsidiary) had over $500M in revenue from their search engine deals...
See "State of Mozilla 2016" https://www.mozilla.org/en-US/foundation/annualreport/2016/ and also check out the Foundation's financial statement and tax form PDFs on the bottom for more details.
The original reason behind them establishing Mozilla Corporation over Mozilla Foundation was that they can keep this practice going while being safe from taxmen reprimanding them for pocketing what is a charity income from tax standpoint.
Now, Mozilla Corporation bills Mozilla Foundation for "service offered at a market price" to do its "socially beneficial, free of charge and any expectation of remuneration" activity, which is selling ads.
Google’s behavior with Android/AOSP suggests they’re more than willing to make the open source version of Chrome useless in practice the second there is no viable competition.
I don’t see any organization other than Mozilla that can keep them honest.