Being a fluent Irish speaker, I agree that there are no soft C's in Irish.
I grew up in Ireland hearing Celtic consistently pronounced as Keltic, the only exception being when people referred to Glasgow Celtic, or Boston Celtic football teams, when the soft C was used.
I have been involved in the Pan Celtic movement since the early 1970's and attend at least three Celtic Festivals each year, mixing over the years with thousands of my fellow Celts from Cornwall (where I now live), Ireland, Scotland, Wales, Isle of Man and Brittany and I have never heard the soft C pronounciation being used by any of the attendees, except in the context of football.
I also did 5 years of Latin in school where our Latin teachers always insisted that we used hard C's ......... so words like Caesar were pronounced Kaesar !
I am impressed by the level of linguistic and historical erudition in the previous posts on this question of pronounciation, but I would remind people that what is actually relevant NOW is that those of us, who are keenly aware of the importance of our Celtic identity and heritage, actually refer to OURSELVES as Celts (with a K!) and it is irrelevant as to what other non-Celts choose to call us.
I accept that the rules of English language pronounciation would indicate that the soft C should be used, as has been trumpeted here by the pedants, but us Celts (with a K!) have historically been a "rebellious lot" and in my opinion COMMON USAGE will always triumph over linguistic pronounciation rules.