" Especially the teachers' pensions, as private schools, and their teachers get a massive leg-up by being able to leech off the state instead of making their own provision."
That's not exactly accurate. The TPS is an unfunded pension scheme, which means it's a typical public sector black hole.
However legislation means private schools are entitled to use it. As of September 2019, private schools will be required to pay 23.6% of teachers' salary, as against the current 16.48%. This increase will not apply to state schools, which will receive money from the government
This increase will add around 4% onto school fees, over and above inflation.
It seems that private schools would be better without the scheme as the government can just revalue and increase their contributions by almost 50% overnight, noting that the average FTSE100 defined benefit scheme has an employer contribution of just 10%.
There will be contractual/union reasons why private schools may need to keep with the TPS, especially as TPS members are required to offer it to all teachers or not at all.
If it were a private sector scheme then they would simply have closed down memberships a decade ago and replaced it with a defined contribution scheme. However it's not, so suddenly the government have said 'we've been doing a shit job of managing this unfunded black hole, gives us more money', and they will end up paying 23.6%. Fairly obviously they would be better off with a defined contribution scheme paying a typical 10%. The pensioner benefits might be worse, but the value to the school of the extra 13.6% isn't there. As they are in the private sector, it's normal to expect a private sector pension scheme, not an underfunded, over-promised government shit show.