So if private insurance wasn't spending €2 billion a year on healthcare consumed by Irish citizens that healthcare would not have to be provided or could be provided by the State funded service at less than the State currently spends?
There are 24000 kids in private schools. The State funds a teacher for every 23 pupils in fee paying schools. It fund a teacher for every 19 pupils in fee paying schools. Therefore the fee paying schools save the State the cost of 120 teachers at the moment. The State does not fund the cost of buildings or associated costs in the 50 fee paying schools. If all of those schools were not fee paying the State would have to provide an extra 120 teachers and the upkeep for 50 schools. The parents of those children have already paid for a place for their child in a public school through their taxes. They are paying twice for the same thing. That's a subsidy.
How is the state, paying for teachers salaries in a private school, a subsidy to the state?
In any case from the top of my head there are over 5,000 schools, 20,000+ teachers. The numbers you cite are chicken feed in the scheme of things.
And again, you expect the state to provide the infrastructure if you need it, but if you choose alternative means of your own free will, you expect some credit in the form of tax deductions?
If I buy a bottle of water, instead of drinking tap water, should I expect taxback?
You are avoiding the obvious dilemma here. You seem to want the option of paying privately for everything and in turn pay no or little taxes if that is the case. But simultaneously, to create such options you will require the organs of the state to provide the necessary infrastructure.
For instance, this discussion, in this format, is only possible by virtue of mobile and internet technology providers using the states communications apparatus. For that we all pay taxes. Even those who don't use the internet. They are subsidising you and me.
In the end, you would need to micro-manage everyones affairs to gauge at what point they use state services and at what point they pay privately, to such an extent that it would be the nanny-state in extreme.