Google didn't pay Mozilla to add a search bar. Mozilla added a search bar because that's a sensible feature, Google just paid to be the default.

If the search bar didn't exist and Google paid to be the home page, fewer people would find out about alternative search engines and id switching was more effor that changing one setting, fewer people would do it.

I just opened the AI sidebar for the first time and it gave me a list of 5 options, along with a link to a help page that compares them and links to each one's privacy policy. This is 1000x better than the current way people use AI, which is to bookmark ChatGPT and never try anything else (well, unless Gemini is shoved down their throat).