> but if an LLM provider engages a search engine to get "live data," that data isn't live but previously crawled as well

I'm not sure frontier labs do it, but imo fetching live data requires fetching both the search engine (1) and the website/page (2). The search engine gives you search results + content snippets (potentially stale), the website/page gives you the actual content fresh from the source.

The tradeoff I see here is between liveness/staleness and cost (hitting an index if of course cheaper and faster than querying live websites again).