I ran two classical stations before 1970, one in Ecuador and one in Richmond, VA (for EZ Communications). Both played most material off records. In Richmond, we had a library of performances distributed on tape as well; the advantage being that a whole piece could be on one reel rather than across both sides of one or even two LP recordings. This allowed for a complete work to be run just as performed.
While someone at WCAO (600 AM) might have done the meter readings for the FM, I doubt they could have dealt with the complexities of changing reels and cuing them in the time that a 60's era two-minute song lasted. So I think its likely they had a board operator.
As an observation, some classical stations had board ops much of the time and had one or two announcers who did the presentations... even back when the voice tracks were on reel tape, not carts. The reason was that it was hard to find voice talent who could accurately pronounce the titles, composers and orchestra conductor names, and listeners were not tolerant of mistakes. So they voice tracked some or most of the day with real pros who could also comment from the recording liner notes, as well as HWV, BWV and other classification catalog numbers.