You have no idea how funding works. Firstly the service is provided by education therefore, it will be 0% NHS funded. The parents haven't opted out of the NHS, but not opting out of the NHS doesn't affect education funding.
The central government get all the tax payer's money. From this they allocate a budget for Education. A sizable proportion of this budget is then divided up to individual Local Authorities, based on the following criteria:
- A amount per child
- B amount per child with SEN
- C amount to cover the support for those children with EHCP plans.
The LA will then keep a little of this money to fund services it provides to support the work of all schools, for example that to cover funded transport, or to cover LA advisors to help schools, or clinics to support children with particular needs.
The larger amount will divided out to each school, allocated using the above criteria, A & B to form the each schools' general budget and C to enable schools to pay for particular support for their highest needs children.
This clinic, yes will be tax payer funded in that it will be funded by Government Money. However NO money will be there to fund it for this child because when the LA was funded for the services it needs to provide, this child was NOT counted in 1 or 2 or 3, because they are not in LA education.
Taxes, including those paid by the child's parents, will go partly into education funding, but the fact that these parents have removed their child from LA education will mean that the LA receives less funding than it would have done if their child wasn't removed. So yes while they still pay tax, removing their child means less money for the LA, which means less money for services for that LA. It makes sense that the child who's parents decision has meant less money is spent on education feel the pinch of less money being spent of education.