of course it does.
Why do you think google buys the rights to firefox's search bar (as a default setting)?
That's not remotely the same? A default setting that can easily be changed for a feature the vendor didn't have a solution for?
To give you an example. Try to use Google Search without sending your data to Google. You cannot use the product without it, you cannot opt out. Firefox, you can use just fine with Google not being your search engine.
Don’t they buy the search bar to have another competitor and not get forced to give away chrome for antitrust reasons? I don’t think they care about the search bar THAT much, it’s basically a donation right?
And by extension, all users of FOSS must be the product, right?