The reason why it has to be this way is that exam board staff do not work for free. The school, as the exam centre hosting and sponsoring the candidates, arranges and pays the board for the exams.
You have opted out of State education in its entirety, so the exam fees are passed to you. That opt-out is a binary for practical and principled reasons.
The most obvious principled reason is: State schools usually only offer GCSEs, whereas private can offer any leaving exam they like, including useless ones that prove nothing as well as highly-regarded exams like IB.
- If the Govt funded GCSEs for private pupils, but not other leaving exam types that may cost a different fee, that would impose a two-tier system onto private schools. You'd have the parents of IB kids starting threads saying "the GCSE kids in my private school get their £750 exam fees paid by Govt, why don't my kids get their £996 IB fees paid?".
- If the Govt agreed to pay £750 towards all private pupils' leaving exams, they'd risk paying taxpayers' money to dodgy overseas boards for diplomas best used to clean one's backside.
A line has to be drawn somewhere about how much the Govt will pay for exam fees, to which companies, and for which children. The State is reasonable in saying "£750, to British exam boards for our home-grown GCSE exams that we have some trust in, for all children who we are already responsible for educating".
You were offered free education for your children. You chose to decline that offer, so you chose to take both the opportunities and costs associated with that choice.