It a false and over simplistic argument to say you are saving the country money by sending your kids private.
The best thing for the country would be for all children to be educated together in a well funded system and well equipped schools. The most successful countries on earth either ban private schools or have very little private school use (see Nordic countries). What the country needs is whole generations of children who have had an excellent education, of course with options to branch off in different directions mid teens, not a few who have had the chance to purchase an advantage possessing this excellent education.
Because rich / better off people have the chance to opt out of the system they consistently vote for low tax parties and not for parties that would increase tax to properly fund schools. We have a problem with school underfunding leading to badly educated adults, and it’s a huge problem in his country. Allowing those who have often had advantages themselves to opt out of the national education system for their children means they have no interest in voting for parties who would solve this problem as they don’t care.
Allowing a few to purchase an advantage is also a detriment to our industries because it skews the job market later on in favour of some adults who are not the most able individuals (on an even playing field). This is also of course why some want to purchase private education.
You therefore acting to the detriment of the country by paying for private school for your kids. You aren’t saving it money or doing it a favour. You are a problem.
The government could ban private education as in some Nordic countries. This has worked well for them. It gives everyone a stake in the system. Short of doing this, it should not incentivise this detrimentally behaviour by giving it tax breaks. They are tax breaks.
As I said, if anything it should be disincentivised with higher taxes like cigarettes and alcohol, which also drain the overall economy as they make people sick.