There's a little bit of truth to it. As far as I know, Mozilla is not actually directed to do anything by Google in exchange for the money. I'm sure if they had too many missteps that Google would begin reducing their funding, but in general they don't have profit as a driving factor in development and Chrome still does.
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.