Users absolutely don't care where their browser gets paid, but financially, Mozilla Corp is dependent on an external entity's good graces to pay them.
Mozilla Corp makes 400 million per year from Google money. If this money dries up in 2023, then the browser has to find a new deal, or close shop (figuratively speaking; I'm sure it'd lumber on since it's an open source browser). This is a fair amount of business risk, so "saving" the browser probably would involve figuring out how to keep the lights on without a search engine deal.