The majority (perhaps the vast majority) of parents would pay for education privately if there was zero state provision. This is the case in parts of the developing world where state capacity is weak. The point of universal education free at point of use is to ensure that children with incompetent parents get an education too.
You can also make the point that taxes are simply a repayment of the loan of a free education you got as a child.
I get deeply ambivalent about this debate in Ireland. First, nearly all schools are private in the sense that they are owned by a body other than the state. But, then all schools are also public as somewhere between 50% and 100%* of their capitation plus capital plus teachers salaries come from the state. I know a lot of people who went to fee-paying schools (I didn’t) and I cannot manage much of a distaste for them - indeed many are my friends!
I can’t see much difference in terms of educational outcomes (evidence does not support this either). The only difference I see is that they still tend to hang around together 20 years later. Whether this results in better labour market outcomes is moot. There are lots of networks in Ireland. Former inter-county players tend to find jobs easily enough for example.
You could take an ideological position and remove capitation from fee-paying schools. I would hazard that of the 50 or so the bulk would opt in to the state system (indeed some have done so in recent years). You would be left with about a dozen who would go full fee-paying, nearly all in or around dublin. This would mean a doubling of fees, probably a reduction in numbers and more exclusivity than at present. These schools would probably innovate a bit more, like lecture-type classes for some subjects and maybe start teaching the international bacculaureate.
*John Scottus is the exception I believe since Aravon closed. But am open to correction on this.