The issue with exempting SEN (which should be the very least that happens) is where are the lines drawn?
Currently exemption is an EHCP with the LA naming and funding the school.
Lots of children don't have EHCPs - especially in private schools - because parents have decided their time and money is better spent finding appropriate provision rather than fighting the system for years.
I was very much dissuaded from applying for one for my child - and by the time I realised it would actually be very worthwhile to have, it was too late because she would have left school before we got to the end of the process.
Do you set it as having exam access arrangements? A diagnosis?
We paid privately for DD's ADHD diagnosis as it's a 3 year wait for assessment where I live. The GP said he was happy to refer, but given it was a case of when she got the Dx, not 'if', then if we could he encouraged us to pay and get her medicated fast. So if you need a diagnosis, you are prioritising those who can afford to fast track that.
Lots of disabilities are on a spectrum - both DH and DD are dyslexic. DH makes the odd typo, DD struggles massively to write a sentence (without tech).
ASD is everything from a bit quirky to non-verbal and profound disability.
What about depression, anxiety, social phobias, school refusers?
If you save 20% on fees, that incentivises a lot of parents without SEN children to find something that ticks the boxes. You now have vast numbers of PS kids in the diagnostic channels meaning longer wait times and fewer resources for those in state or HE.
Therefore in the interests of saving councils, NHS and teachers vasts amounts of time, effort and/or money... it is considerably easier just to not have VAT at all.