But surely, somebody would pronounce Brighid /brIgId/. The h after the 'g' stopping the sound from affricating.

Certainly, but in the old spelling, the h would have been represented by a dot (séimhiú) over the "g", which I can't reproduce here, so I wrote it as a "h". It would really have appeared as "Brigid" with a dot over the "g", and in practice, the dot was often left out.

I wonder if there is any connection with the Germanic names "Birgit" and "Birgitta", or if they are from a different origin?