I don't see search and AI as fundamentally distinct things. Usually I just want an answer.

Maybe we use search differently, but I very often don't just want an answer, I want to find a website to help me. Maybe it is because I need to do business with a company and need to find their website to interact with them, or maybe I saw a cool site awhile ago that's relevant to what I'm doing now and didn't bookmark it (because I dropped that habit when Google search was good), or want to read the official documentation about a product I bought, that someone already put a lot of effort into making complete enough and digestible to a wide audience... and the LLM responses tend to get in the way.

Like the parent I use good/paid AI when I want an AI response. So, yeah, an omnibox that knows when I want "an answer" and one that knows when I want to find a thing sounds slightly more convenient than switching between two tools, but Google search is not that Omnibox.

If you don't care about the facticity of the answer, AI is less clicks, granted.

For the same reason I read a book instead of just the plot summary on the back cover