A search powered by sats instead of advertisements (like DVMs) won't be a default in Firefox, or likely even considered, while there is funding from Google. Mozilla has explored other options, but I still think Google is the largest source. It is certainly influential to development.
Discussion
There is no guarantee your search is not being tracked unless you take active adversarial measures.
Changing the business model from ads to pay-per-query might improve the incentives, but it offers no guarantees.