I'm pretty sure they can, and do. The vast majority of Mozillas revenues come from Google for being the default search provider. The fact that Google pays Mozilla, Apple and Samsung (among others) to be the default search provider has been an issue with regulators, but as far as I know there has been no rulings on it (yet?).
Stage 1: Buy Chrome from Google, with its 65% browser market share.
Stage 2: Tell Google you'll keep them as default search provider for $5 billion per year.
Stage 3: Profit