You could also use the responses api which stores all message contents (including reasoning) on OAI servers. This has been possible for quite a while now. Encryption is only necessary if you really care about local storage (which is different from privacy concerns, because the data gets sent to their servers anyway).
well the encryption part is also mostly about OAI wanting to avoid others to distill from their COT/reasoning traces, since this is not ever displayed to devs or final users, and as you say lives on their servers.
but yes you're correct on the responses api already baking it in too
supposedly keeping these between tool calls should help the model reason and have better overall outputs etc