The argument being made against taxing private schools is that a small minority of students from private schools might move to the state sector due to rising fees. However, given that most private school students come from the wealthiest households (top 10%), this issue shouldn't really exist.
Wealthy families should be able to afford the tax increase and if they can't, then it raises a bigger question: why aren't the wealthiest people paying their fair share of taxes in the first place? Instead of worrying about a small percentage of students moving to state schools, the focus should be on ensuring the wealthy contribute fairly to public services, including education.
The truly wealthy in the UK often avoid or minimise taxes through legal loopholes, so expecting them to pay VAT on private school fees isn’t unreasonable, it’s simply making the system fairer.
Sources: https://www.theguardian.com/education/2019/jan/13/public-schools-david-kynaston-francis-green-engines-of-privilege and https://www.llakes.ac.uk/wp-content/uploads/2021/03/RP-62.-Green-Anders-Henderson-Henseke.pdf