Google already owned search when firefox finally had nontrivial market share. They did not "buy their way into search", the transformed the market with an excellent product (Google altavista, the undisputed king until Google came).

At some point, they started paying e.g. mozilla to maintain their position.

But they reached that position based on merit. (A rather uncommon event, unfortunately).

That has only really bore fruit in the US though. MS have less than 5 percent share of search worldwide.

"Google bought their way into search by paying firefox"

Google actually had a larger share of the search market prior to the Firefox deal.