Irish Republic gets under my skin for the reasons you outline here. Usually in the context of the term being bandied about by a Unionist politician* or a Tory MP. I have a niggling suspicion it's used as a passive slur similar to the way Russians habitually stick a "The" in front of Ukraine ...like it's a homestead or a reservation.
*in an interview with RTE radio a while back a DUP representative (I think it was Mervyn Storey) pulled up the interviewer for referring to his jurisdiction as "The North" and insisting she call it "Northern Ireland". Yer wan apologised, shrugged it off and got on with it only for the DUP guy to go off on one with multiple references to this mythical "Irish Republic". I half hoped she might ask him to watch his Ps and Qs while he was at it but in an interview situation...
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