I don't think thats a proper comparison. In your analogy of the NHL and sports networks, the league would be the equivalent to EP's/Varsity and the Fox Sports/Sportsnet would be Flo. Both the league and the networks have agreements that claim rights to content, especially for marketing (which would include YouTube clips posted to network and league channels).
Gyms, in this analogy, would be the person who is not allowed to record at the game or the person who records the televised event and uploads it to YouTube. They are not part of the agreement between the service and the provider. Gyms are the consumer and have no rights here. (EDIT: Or you could argue the gyms are the hockey players on the teams, who still have little to no rights to content.)
Leagues and networks go after people on Youtube, Twitter, other media outlets, etc all the time. It does happen in other sports, it happens everywhere.
As much as I adore the idea of gyms having their own media channels so their fans can cheer on teams rather than recording, I'm doubtful Flo is going to reconsider their policy. Flo has nothing to gain and everything to lose from allowing that.