A lot of the stuff you wrote about is not what most of us are talking about when we call it intellectual catfishing or say your intellectual fly is open. For example, I have few problems with people using LLMs in private for purely personal things. I still won't go ahead and recommend it, but it's a decision we can all make for ourselves. Also, writing an email to your city is something you could ask your assistant to do if you were super rich, and using the LLM to do it isn't that different, so I'm not going to judge you for that.
As a more nuanced case, someone could dictate a long rambling stream of thoughts full of contradictions to an LLM to transcribe and summarize, and then they could reflect on the summary and write the real thing themselves. That's like talking to someone about a subject and writing about it afterwards, and also not something I would really take issue with.
That's my partial answer to the cognitive prosthesis question: keep it off to the side as a tool to help you do the work. There are still many caveats here, and the more complex the demands you make of the LLM, the greater the risk of some kind of break down. For example, continuing the example in the previous paragraph, if I did have the rambling conversation with an actual person who helped me refine and clarify my thoughts, I have some kind of mental model for the dialogue partner that can help me correct for some biases. With an LLM, anything resembling a mental model I could have would be very far off the mark, so I would need to be aware that I can't treat its output the same way I would treat something written by a person.
(By the way, for myself I have a stricter rule: I would only use an LLM in a way that made me a better person independently of the LLM and not dependent on the LLM, and thus I don't use them at all. That's what I actually recommend, but I don't expect everyone to adopt that rule.)
My main problem with having the LLM just do your writing is that it's simply a misrepresentation. It's wrong to claim you wrote something unless you chose the words. I really empathize with the struggle of expressing oneself clearly, but there's just no getting around this. And to be clear this isn't something pedantic or just a technicality, since the act of formulating a thought in an actual human language with valid syntax does require a level of care and attention that simply is not there unless you do it yourself.