My take on it is that software able agree it did something wrong

This would make sense if it made an independent subjective assessment and arrived at the same assessment of the user. But that did not occur, it simply incorporated the user's subjective assessment as a new piece of the narrative and continued from there. If the user had expressed significant satisfaction that it took care of that email all on its own, it would have also agreed with that. In circumstances like this, the LLMs are generally set to agree with the operator in any subjective matters.

One could imagine the context having something like "before taking any action, if the user expresses dismay at the result and this would result in an agreement with user on poor assessment, then don't do it", but then if that were actually honored, then you pretty much disabled most all actions, as the llms are going to agree with any poor assessment.

Well, it still doesn't know it was doing anything wrong.

Most of the LLMs are tuned towards agreeing with the user, if you had a LLM able to stop a train crash. And you then blamed it for stopping the train crash, it would still start with a "sorry"

It's an LLM. All it's supposed to do is generate output that fits with the given input. There isn't some logic in it that makes sending an email an error, or anything that enforces detecting any kind of error. You can wrap it in other tools for that, but there isn't any indication that's the case.

More than that, the initial post says the user found out about the email and then queried the LLM, so it didn't detect anything. There isn't any bug here, this is how LLMs operate.

Edit: here's a podcast discussing exactly this: https://youtu.be/RoIjWGi-w5o?is=2ly14A2UdKalHuAH