These things are not mutually exclusive. Allow me to specify a city or state or county or country or zip code as a bang in my search and show me good results based on that. Both problems immediately solved. I wouldn’t be any more or less reassured about a search engine’s privacy stance if that feature was offered to me. This is a feature I can absolutely use in a private way (I can do that search over tor or a vpn with two hops if I so desire), and it gives me the control over what I provide the search engine and how and when.

Right now search engines don’t provide an interface for good location aware searches that you can manually specify - you have to let them build a shadow profile on you via all sorts of privacy violating fingerprints or just give up location aware searches altogether. There’s no reason it has to be that way though.