That number is true, but covers a lot of tiny religious schools - it's 75% of schools represented by the Independent Schools Council which is what we'd normally think of as a private school educating a body of pupils, where you'd have reasonable expectations of social responsibility in line with charitable status.
There is zero risk in removing this status, on the contrary it is votes in the bank. The private school sector has been on the take for years in this department (in aggregate, there are some impressive exceptions), trousering tax relief on new playing fields whilst failing to widen access to bright kids from disadvantaged backgrounds. UK voters of any stripe dislike this sort of brazen chiselling of the system, and you can times that by ten when it comes to kid's education so no idea why you'd see that as a high risk strategy. Just need to shine some sunlight onto it.
We're not talking about all the money in the world here, but it's a critical reality check and a step towards tackling inequality - hard to get progress on this fundamental issue when we've got all these private schools hanging off our teats.