For the purely pragmatic reason that if the struggling single mum has to use the laundrette, it doesn't cost the government any more.
If the law was that if that single mum couldn't afford the washing machine, the government had to buy her one (albeit a more basic model) - and lots of people could only just afford washing machines - then it probably wouldn't make sense to charge VAT on washing machines either.
If a family chooses state school over private because of VAT, then the government becomes liable for the £4k-£7k per child which state education costs. If the child is in private school, their parents pay the lot.
People say that one more kid joining a class doesn't cost anything, and it's true that marginal cost is likely to be less than average per-child cost. Up to a certain point.
But where up to 30% of students are educated in private schools (eg Surrey, Edinburgh, Bristol) then I don't think you can fudge it on 'one extra pupil doesn't cost anything'.
And that's without the other reasons like increasing the education of a population being a public benefit (even more so than clean laundry) and that private schools allow parents to work more due to extended hours which grows the economy and increases income tax revenue.
No one really agrees at what point adding VAT to school fees starts costing the government rather than raising money due to having to educate more children. The main number I've seen is 10% of children who would have gone private instead using state school.
No one really knows how many children will use state school instead of private as a direct result of this tax. Personally, I think it's likely to be more than 10%. Note that whilst children switching is a short-term pain point, what matters for long term costs is how many families simply don't start private school due to the higher cost.