I'd take out Eire from your comparision (or more correctly the Republic of Ireland as you are writing in English not Gaelic). The RoI had a parliamentary democracy as we do in the UK but with an elected upper house rather than appointed and an elected head of state rather than hereditary. The President of Ireland can no more effect policy or run the country than can the Queen. They sign off acts of parliament, shake hands, open hospital wings, supports charities, represents Ireland abroad and all the sort of stuff the Queen does but at a fraction of the cost.