I agree that the campus writing center should be on every freshman’s list. Go early and get to know the tutors.
I worked in a campus writing center for a couple of years. Too many students come in at the last minute. They should have come in much earlier. Stating the obvious, but I tell all my students that when a professor gives you a deadline for three weeks in the future, it’s because a student needs to spend three weeks working on the project. Not one week, or worse, two days.
I had students come in who had assignments due that afternoon. I didn’t judge them, but again, it goes without saying that the student was lucky if they got a passing grade.
The level of writing expected at college is higher than what they did in high school. It’s even more important to use the right source materials and to learn how to do citations. I tell students that citations are basically how you prove what you state. My favorite source for all things writing related is Purdue OWL. Here’s the link to their citation styles page: MLA Formatting and Style Guide - Purdue OWL® - Purdue University
To the OP, if your child is at all inclined to spend some time prepping for college writing, have him or her look at this page, if nothing else. Citations are time consuming and confusing. My #1 top tip to make life easier in college writing is to note WHERE your source is when you cite it. Do it immediately because it’s very hard to go back and find it later.
Of course, the most basic and critical rule of good writing is support your thesis. I always ask the student to tell me if the sentence or paragraph in question is supporting the thesis. If it isn’t, fix it. Or scrap it. The thesis is pretty much always in the first paragraph and I tell students they should refer back to it regularly. It’s pretty easy to drift away from a thesis statement, so the student needs to remember where they were going with the point in question.
I wouldn’t worry too much. My son, who is dyslexic, was a terrible writer. Unfortunately for him, he had to run all his high school writing by me. It was painful for sure. In college, he has come to me and asked me to review his work. His writing skills are decent now, but he’ll never be an editor, that’s for sure. He has me so he doesn’t need to use the campus tutoring center, but the kids who work there know what they are doing. They will be trained and have good writing skills.
And to bring this back to the OP’s question, I don’t see any benefit right now to have them tutored for college level writing. Parents have approached me with similar requests. Without an assignment or particular goal to aim for, tutoring isn’t going to help much. It’s just going to frustrate them and turn them against writing.