Adam:
I don't know if Emmet had a relationship with James Braid -- I'd be surprised. But what makes you think that Braid was the one behind the hole names at Gleneagles? I am trying to recall if he named holes on his other courses -- I don't remember any from St. Enodoc or Pennard, but I'll try to remember to go back and look at their scorecards.
It's possible that Emmet knew someone else who helped name the holes -- perhaps C.K. Hutchison, or knowing Emmet, perhaps the people who owned Gleneagles!