It doesn't seem to have any internal tools it can use. For example, web search; It just runs curl in the terminal. Compared to Gemini CLI that's rough but it does handle pasting much better... Maybe I'm just using both wrong...

It does have web search - it's just not enabled by default. You can enable it with --search or in the config, then it can absolutely search, for example finding manuals/algorithms.

Use --search option when you start codex

web search too is off by default