Nice post. Good for discussion for my A-level Economists on a range of issues.
As a default libertarian who nonetheless agrees that the NHS is the least inefficient way to solve healthcare market failures (and a teacher in an Outstanding state secondary in an affluent area that gets ‘community outreach assistance’ from local private schools — unnecessary, poorly-targeted but welcome), I would suggest that the Charity Commission should beef up its criteria:
Any private school that offers assistance to its nearest school (at the same key stages) that has above regional median percentage of Pupil Premium students
The requirement is that the PP gap falls and Progress 8 score rises, over a three year moving average (measurements to start after one full cycle of the KS) in order for the private school to retain their charitable status. Otherwise, pay the VAT. (N.B. All the above thresholds subject to someone with better data fine tuning them)
This allows private schools to make a choice on how to use (up to 20% of) their resources according to local need (e.g. pay for FSM, TAs or giving more hands-on help) and only rewards results.
If the private schools aren’t up to the job, take the VAT and channel it to schools that did not get effective support.