The first principle of the state is the promotion of the common good. So tax-funding of education isn't wrong in principle.
Practically, the exemption of childless families from education funding will only increase the rate of childlessness, and hence the extinction of society.
By the logic you use, that is correct. However, I would argue that the first principle of the state is to protect it's citizenry and to uphold justice. (Perhaps that fits under your definition.) In any case our public school system is so poor that I am unsure if public education qualifies anymore as the "common good."
My real argument, however, is that the civil government as designed by God (Romans 13, Pentateuch etc.) was never really intended to be the primary educator of the people. Like social welfare, it is not fit for it and doesn't do it well.
'The first principle of the state is the promotion of the common good. So tax-funding of education isn't wrong in principle.
Practically, the exemption of childless families from education funding will only increase the rate of childlessness, and hence the extinction of society.'
That sounds very Marxist to me. The purpose of government is not to perpetuate itself or the society it serves. The purpose is to protect its citizens' right to live their life as they wish. To that end, it may enforce contractual obligations amongst the people and it protects the people from the use of force.
With regard to education, you are saying that people should not be discouraged from having children just because they can't afford to educate them. This is no different than saying people should not be discouraged just because they can't afford to feed, shelter, or clothe children. I think those are excellent reasons for people to be discouraged from having kids. It is irresponsible to have children when you know they will be a burden on your neighbors.
As far as excluding childless families from paying school taxes, the problem is really one of timing. Paying taxes over a 60 year adult life allows people to AVERAGE the cost of educating their children. You cannot skip paying taxes in your twenties just because you don't happen to have children until your thirties and they are finished with school by the time you hit your fifties. With careful planning, of course, you could save "early" for your kids' private school education before they enter school age, and borrow money which will be paid back later in life. Many people do that.
However, if taxes are going to support schools, then you must be forced to do the "early" part via taxes, and also taxed later to repay the "borrowing". Funding education via taxes means the public purse cannot allow people to avoid contributing on the promise that they will remain childless. The state is applying force based on a pre-emptive assumption that you may not remain childless and somebody will have to pay to educate those children. It is almost as though you are assumed guilty and your promise cannot be trusted.
My take on education is this: The state should only REQUIRE that children receive an education, and TEST to make sure it has been accomplished. This amounts to an unfunded mandate on parents, but that is where the burden should be anyway. This falls within the state's proper role of protecting its people -- even from irresponsible parents who might otherwise cripple their children by not providing an education. It should OFFER a public education, but if it cannot do so competitively with the private sector, then it should not attempt it. It certainly should not have the 'fabricated' cost advantage over private schools that it now enjoys. The state option is one which parents must pay for whether they use it or not, and the alernative requires large life sacrifices. That is NOT a level playing field and the result is the near-MONOPOLY the state currently has on education.
Vouchers could at least provide some leveling and force the state to compete for its 'customers'.