>"to compete with Google and the likes, the scale, cost and complexity would be huge."

CloudFlare's entire business is scale, cost, and complexity. They are powering like half the web at this point. Wouldn't really call them a "new entrant".

Codex and Claude Code need to do countless web searches, I'd guess they have a partnership with Google. Open models don't have this partnership so the search API needs to come from somewhere.