Sorry don't agree with private sector paying for free school diners for state educated children .My partner paid for private schooling for his son.
He was not in a highly paid profession, although he was in a position of responsibility. I chose to send my children to state school. My partner at the time lived in a very modest house in a area that I would not want to live in, he made many many sacrifices for his son's future and had to endure many unkind remarks from neibours who thought he was loaded or snobbish. He was neither, just a Father from the West Indies who wanted to give his son a good start in life knowing racism still exists in the job market and he wanted his son to be well educated and have the best. His son now has a wonderful career and a good future. The school's where he lived were not good.
As I mentioned above I chose to send my two children to a state school, but I would not expect the private sector to pay VAT so that my children could get a free meal, I would take responsibility for that. I would also add if they had added VAT my partner would not be able to send his son to that school.