They don't care. This is purely for a business upper hand.

OpenAI should probably encrypt the chats and lock itself out going forward. Collect whatever metrics they need on the fly before locking.

Is there an expectation of privacy using ChatGPT? Do users think nobody is ever going to be looking at their logs?

I find it interesting you are blaming the NYT on this and not ChatGPT for keeping these logs in the first place. If openAI didn't keep logs, then there would be nothing to search, and a more harmful actor couldn't accomplish something far more nefarious. Saying that there could be confidential information in the logs, so that means we shouldn't access it, should also mean the logs shouldn't be kept.