BlameGame
Secondary schooling in Ireland was not always free, until the 60s all schools charged fees. When Donagh O’Malley introduced free secondary education some schools opted in to the free scheme and got higher levels of subvention than those that did not. The system has functioned in this way for the past 50 years and works pretty well: the teachers are provided by the department for all schools, so there is equality in teaching; free schools have all support staff and building paid for by the state, while those that charge fees provide there own support staff (admin, grounds, etc) and are responsible for their own capital building programme. Withdrawing all financial support from fee-charging schools is not just a thing to do on a whim, it represents a total re-drawing of the education system as it has been for half a century. It will be very costly and I have yet to see any sort of convincing argument why it would be a good thing for the country.
The reality is that if you live in Dublin and want to send your kids to a school that is co-ed, non catholic, and with a good infrastructure and academic reputation, your options are severely limited. I’d love to be sending the kids to a free school, I don’t pay out all that money because I love spending money. The sort of school I pay for is pretty standard in most countries - the all weather pitch and the science labs and the other facilities are what you’d see in a decent middle class american high school or State school in the UK.
Two canards to nail
Myth 1. all fee-charging schools are elitist and segregationist
This is the results of a survey of Protestant fee-charging schools taken last year, if you look at the income spread of parents you will see that the kids there mix with children from all financial backgrounds. You will also see that the main reasons that people choose these schools are because of things like ethos, mixed gender, religion, and subject choice. Friends and family going there and sport are bottom in the list. People may prefer to believe that all kids are Ross OCarroll-Kelly but that is a stereotype, a caricature.
Myth 2. We are the only country that does this
France has a similar system. France with its Liberte, Equalite and Fraternite. There may be others, in fact it seems a lot more likely to assume that there are others like it when you can show two countries having evolved such a system. On examination of Ireland and its two nearest neighbours (UK and France) you could say that a majority 67% have such a system of part-funded education 
And as for the ‘carve out your taxes’ argument: why shouldn’t I? Really, I pay out over half my income in taxes, the only thing that I get in return is education for my kids and (touch wood) a very limited experience with the health system. If I choose to top up what the government provides to the school, from my taxes, to pay the teachers, why the hell shouldn’t I?