This is getting silly. I thought it would have been pretty clear by now that there are as many people who Englished their name as O'Ryan as Ryan, O'Reilly as Riley, Connell as O'Connell etc. If there is a rule in Irish, it didn't get consistently applied, and as there was a distinct hiatus in the speaking of the language by most of the Irish population, quite possibly the rule is recent.

BTW I suspect the form O'Bourke is a sort of back- formation; Burke is a Hiberno-Norman family name, and not a patronymic at all, and the form only seems to be common in Australia. The Irish form IIRC is de Burca.

Perhaps the use of O' in English (and more particularly American) reflected the opposing pressures on immigrants to assimilate or to identify with their ethnic origins.

So Carolan/ O'Carolan? Just play the music.