They do not get taxpayer support in the sense that they are noi funded by the taxpayer - any more than my ISA is funded by the taxpayer (because the interest in it is tax free). You could describe it as taxpayer support, but in common parlance peoplme don't because it is misleading.
It is very much overstating the case to say charitable schools are funded by the taxpayer - just as it would be to say for example that RNLI is taxpayer funded (when it famously receives diddly squat despite being an emergency service, but does use all the tax exemptions which apply).
But I agree, tax law and the various tax statuses available are not always easy to understand. So I hope this clarification has helped. There are also a number of experts posting on the other thread about this who explain this far better than I do, so if it's still not clear why it's wrong to describe it as taxpayer support, it might be worth reading them as well.
(TBW, the 'value' of tax not collected from those schools which are charities is about £200 per term per pupil. I think it's very easy to misunderstand what both what charitable status means and what it is worth in cash terms).
Those independent schools which do receive taxpayer support in the ordinary sense of the term are a very small number of very specialist schools. If you do not want to see taxpayer support to private schools, then those would be the ones that would go.
That's not something as would want to see. But it does seem to be a widely supported position (on MN, at least)