Cool may be thought of as primarily a W. Coast thing, with Contemporary, Pacific and the like, but some of my favorite Shorty Rogers sides are on RCA and Atlantic. And Miles – Blue Moods on Debut has to one of the more non-W.Coast cool school records out there.
W. Coast, as Giola looks at it covers the waterfront from Kenton to Mingus & Dolphy, The lighthouse crew, Brubeck and Desmond, Chet & Mulligan, Dexter, Hampton Hawes, Giuffre, Shorty, Shelly and even Ornette iirc.
So to answer your original question, no not really, but maybe sort of.