He hadn't seen the blog I wrote on the 13th November entitled Gilmagu - Carthagus so I sent him a copy though pointing out that it wasn't written to address his etymological question above. I said I could see how one might be tempted to see a link from the Gaelic Carthach via the Latin Carthagus via the English Carthage to '-thage', to '(McIl)hage' ..... However my comment must be that there is no form of paper trail which supports such links, and we have the much more obvious evolution from Carthagus to (Mac)Carthy. Why would one drop the first syllable?
I admitted that I don't know anything about Celtic mutations and that in thinking about all this I have relied heavily on those who have the status of 'experts' in the surname field, not least G.F. Black, and I think they have been right to quote what I have called 'the paper trail'. I attempted to summarise the evolution as follows: the Gaelic Mo chutu became Mochuda became the Latin (Gil)magu became (Gil)mahgou [became (Gil)malgon] became the English (Mc)(Il) ha go became (Mc)(Il) ha ga became McIlhagga. The evolution was of course over many years, perhaps over 54 or 55 generations!