By actual writing I simply mean finding the right words, with the right spelling, a good flow, and well constructed sentenced.
I found LLMs to be awesome at this job, which make sense, they are language models before being knowledge models. Their prose is not very exciting, but the more formal the document, the better they are, and essays are quite formal. You can even give it some context, something like: "Write an essay for the class of X that will have an A+ grade".
The idea is to let the LLM do the phrasing, but you take care of the facts (checking primary sources, etc...) and general direction. It is known that LLMs sometimes get their facts wrong, but their spelling and grammar is usually excellent.