gpt-5.3 was not accessible via API, at least for me

But it was in codex

Ah ha, I know what you're hitting.

5.3-codex is only available via the Responses API, not the Completions API. Two different APIs for model access. If you were using Completions you have to port to Responses. It's not that hard. I did this for my own agent the other week. I think it might be like that for all their new models from now on. Responses is a much more powerful API. It's more like a front to ChatGPT than the underlying models.

I am using responses API (these days I only use responses API when talking to OpenAI)

I'll try again, maybe I was doing something else wrong, or something changed

OK, in that case I'm not sure. But Codex can't hide anything. I don't think there are special codes compiled into the binaries or anything like that.