The issue is not with firefox. The issue is with the search engine you're using. Search engines are different to browsers.

This is what I do as well but with Floorp. What's the issue with Brave Searches AI results? It's never been intrusive nor in the way, and depending on what I've searched for it can be very helpful and save me time. 1/2 of the time it's a button for if I want to generate an AI search.

But yea this isn't related to Firefox.