In my view the nature of the sport had little to do with the failure of the BCA to get USOC (now USOPC) recognition for cue sports in the US. Cue sports were already recognized by the IOC as a legitimate, internationally organized sport. I think a proper application that was rejected by the USOC would have become a discussion between the IOC and the WCBS.
So far as I know, when the BCA saw what it would have to do to be accepted by the USOC as the governing body for cue sports in the US, they abandoned the attempt at recognition. Among other things, I believe the USOC required that 30% of the board of the BCA to be athletes. There was also the problem that snooker and carom should be represented/supported by the same national governing body. It has been a long time since the BCA had much interest in the other two disciplines.