> My question is: what's the point of the AI in the search engine if it itself isn't going to use the search engine first before answering?

Because the point of AI slop is to waste your time. You just lost about 30 seconds of your life trying to get a correct answer. AI was lying to you, so you had to spend time to counter the AI slop lies here.

I solved it by banning all AI slopness; in the browser some extensions do that. The world becomes better without AI slopness.