I think if you asked the average Joe public, they would immediately associate all SAR with the RAF, regardless of the colour of the cab. In other words, the RAF get good
PR whoever do the job. Granted that those who live on the coast might be better informed, but most of the population live inland.
I have had the pleasure of flying with both Civvy and ex mil pilots in a SAR role and IMHO the Civvy commanders are every bit as good as the ex military types.