I don't think it's "only possible" for sure, as maybe everyone was sweatily touching the same door handles or lockers, or maybe their laundry screwed up and cross-contaminated towels instead of cleaning them, but airborne transmission in an enclosed space certainly seems the most likely.
I also doubt mask-wearing would have prevented that completely, either. 61 cases from one suggests it was a pretty virulent breeding ground and cloth masks aren't completely effective. It might only have reduced it.
It's a difficult question: must some gyms (and probably other businesses) be closed and either bailed out or bankrupted in order to control the virus?