Hosted search API from your LLM's provider is much easier to use, faster and less error prone. Also I believe that LLM may work better with its provider's search api, as it was definitely trained on it. I mean, it is just more convenient to use your provider's search.

And if we talking about LLMs, there is no particular point for search to be absolutely free if you still pay for the LLM. It would have more sense if search AND llm would be completely free.