So there's a market niche for a browser which comes from a sort-of neutral entity, and I think Google recognizes that and that's why they continue to pour money into Mozilla. Mozilla looks like a neutral entity, dedicated to the good, while in fact it's almost a wholly-owned subsidiary of Google.
For example, it took a very long time for any really effective adblock extension available in Chrome, and (this has likely been fixed) for a while even when it blocked ads it actually still loaded them so the ad producers couldn't tell difference.
It was because of technical limitations in the capabilities of chrome extensions. Considering adblock plus has nearly double the #2 most used firefox extensions, and nearly 6x the number of users as the #3 most use firefox extension, it seems like the design for extensions should almost be done specifically with enabling adblock in mind.
I'm no conspiracy theorist and I use chrome, but it seems to me that at least in the above case, Google put their bottom line as an advertising company above what the users wanted or what made the best browser experience. Not that it is too much of a bad thing, but I think saying that Google has the same kind of control over the direction of Firefox is silly.
I'm sure you could find plenty of charities that are nearly entirely funded by Microsoft's Charity branch or something, and I'm sure those charities don't go around badmouthing Microsoft but they don't exactly take orders from them either.
Do you think Google also own every single adsense user then?
Chromium (NOT CHROME) is open source, but it does not fight for freedom or general public interest. It fights for bringing Google even more control and user base. In fact Chrome woulnd't even exist if it wasn't for Mozilla.
THAT is the reason why they still use Firefox. If Firefox globally falls behind 20% global market share, Mozilla won't have much weight when doing things such as forbidding h264 support, pushing true open standards, etc.
Google wants strong competition in spaces where they don't make money (in fact this is how all well-run corporations operate)... so throwing some pocket change at one of their "competitors" in the browser space is very good for their business (it keeps other competitors like Microsoft busy).
Google even mentioned (I remember this from 2003 or so) they want competition in their profitable areas (search) since being a monopoly is a) soul-damaging and b) invites government regulation.
http://en.wikipedia.org/wiki/Google_Energy
http://en.wikipedia.org/wiki/Google.org#Renewable_energy
*: reducing their energy expenditure is obviously not entirely altruistic. Powering data centers is costly. But still...
(Note: I've got no sources for this; just a cursory understanding.)
Does Opera have a default search engine?
With much smaller market share than Firefox, Opera still makes a lot of money from the default search engine.
The game isn't browsers. Google makes no money from Chrome.
Google makes a ton of money from online ads. For that, it doesn't need to have the top browser. Having a great browser is very useful, but winning the browser war would be self-defeating.
Compare to Microsoft, which also makes no money from IE. But Microsoft actually does want to win the browser war, since marginalizing other browsers is good for Windows (like opposing anything cross-platform is).
Google's game is to make money. Not supporting Mozilla, while Mozilla is useful to it, would be counterproductive. Stopping to support Mozilla would help Chrome's marketshare, but hurt Google's income. I expect Google to continue to do the rational thing here as it always has.
Google (nominally) have several mutually aligned interests with Mozilla, namely the Open Web.