TBH it sounds like you're just imagining a very different service than the one openAI operates. You're imagining something where you send an input, the server returns an output - and after that they're out of the equation, and storing the output somewhere is a separate concern that could be left up to the user.
But the service they actually operate is functionally a collaborative document editor - the chat histories are basically rich text docs that you can view, edit, archive, share with others, and which are integrated with various server-side tools. And the document very obviously needs to be stored on the server to do all those things.
But the service they actually operate is functionally a collaborative document editor - the chat histories are basically rich text docs that you can view, edit, archive, share with others, and which are integrated with various server-side tools. And the document very obviously needs to be stored on the server to do all those things.