How interesting that you assume that if all the Irish professionals you know went to private schools, there must be something wrong with the state variety? That, if I may say so, is a very British assumption.
There are not many non-state schools in Ireland, and, certainly until very recently, most people just went to wherever was geographically closest, which means there's educationally, far more of a social mix -- lots of UK viewers seemed puzzled by the TV adaptation of Normal People in which Connell, the child of a cleaner from a locally-notorious 'bad' family went to school with Marianne, the daughter of two wealthy solicitors with a holiday home in Italy and a flat in Dublin. That is much more normal here than in England, where I lived for years.
DH and I are professionals with multiple postgraduate degrees, returned from years abroad and living in a traditionally 'prestigious' old area. Our son goes to the nearest school an Educate Together and the children in his class have parents who are anything from architects and neurology consultants to hotel housekeepers and waiters. I actively prefer that. I don't feel the need to socially engineer his social life and friendships.
I think part of the difference in attitudes to class in Ireland is a different economic history we didn't have an industrial revolution at all, really, and it was a largely agrarian society until quite recently, so the important distinctions were between 'strong' and small farmers, and the landless, and smalltown shopkeepers and the like and of course a history of colonisation removed the Gaelic aristocracy and replaced them with a foreign landowning upper-class.
Which is certainly not to say Ireland is a classless utopia the Ross O'Carroll-Kelly column and books are a satire of a certain kind of Dublin 4 snobbery which does exist but it's less entrenched. And far less anxious. English friends seemed far more socially-anxious about their children's friendships and hobbies.