Probably minimal, it's mostly lower memory use from not having to boot a separate V8 engine with its own GC, like how Electron apps have to boot a separate browser engine. But CPU-wise it's not doing anything interesting on the client.
The neat thing for me is just not needing to setup a Node environment. You can copy the native binary and it should run as-is.
There are various solutions for turning node programs into standalone binaries, by more-or-less including Node within the binary. One I've used before with success is "pkg".
If you're parsing JSON or other serialization formats, I expect that can be nontrivial. Yes, it's dominated by the llm call, but serializing is pure CPU.
Also, ideally your lightweight client logic can run on a small device/server with bounded memory usage. If OpenAI spins up a server for each codex query, the size of that server matters (at scale/cost) so shaving off mb of overhead is worthwhile.