€250,000 is the President of Ireland's salary. That does actually go into his own pocket, whereas as far as I am aware the King does not receive a salary. The cost of running the Office of the President is more than €4.8m. That is of course considerably less than the Sovereign Grant, but it covers the same sort of expenses - staff, office costs, travel etc.

I can't find a figure for the number of engagements the President carries out, but it is probably in the hundreds rather than the 3000 or so carried out by the Royal Family, which as you point out includes those carried out by other working royals as well as the king, so of course the costs are less.

If you like to think of these engagements as "junkets" fair enough, although they seem to be popular with the public. You may wish to replace the monarchy with a republic. but I think you'll find that in many republics the president's role includes similar junkets. I also think you'll find that the state pays for them. Republics also have state funerals and investitures. However I quite understand if you feel there are too many of these junkets and that they're unnecessary.

You say "it's a matter of principle". To me, the principle is that the state should pay for activities carried out on its behalf. However I'll leave it there, as you've clearly made your mind up.