I'm maybe a bit more pro-LLM than some other perspectives. My feeling is that using an LLM to help you generate text is fine, but using it as a replacement for thought is not.
In other words, a good cover letter should explain why you are a good fit for the position. There is an implicit question here, that requires thought. Which of your qualifications match the description in the letter? What accomplishments do you have that will make you stand out from the crowd? What are your goals for your research and teaching? If all you do is prompt ChatGPT with "write me a cover letter," it will not know the answers to those questions, and it will show that you have put no thought into it.
However, if you have put in the legwork to have answers for those questions, then you can use a language model much more effectively than simply saying "write me a cover letter." You can write a much more specific prompt like, "Write a letter that talks about my accomplishments X, Y, and Z, and that fits into University A which is looking for B, C, D. Make sure to emphasize my experience in super-duper-modern-technology and explain how it relates to Prof E's research in FGH." You could feed it a cover letter that you've written highlighting your skills and the job description and ask it to modify your cover letter to emphasize things in the job description. If you use it well, the result will be something that reflects the thought you have already put into answering the key questions for yourself.
Of course, once you get text from an LLM, you should not simply copy/paste it and call it done. You should read what it has written carefully, and use it for inspiration. Rewrite sentences to put them in your words (you may emphasize some ideas in a subtly different way), correct errors, add pieces that it has missed or that you have realized are important. In the end, you are still responsible for what you submit, and the text should reflect your thought and your work. But you can certainly use the LLM as an assistant to help you express your ideas.
On some level, the thought is the hard part, and coming up with text perhaps "should" be easy once you have done all that legwork. But you might have writer's block, or English may not be your first language, or just want some help since editing is often easier than starting with a blank page.