Okay, but it's still just an organizational policy, not a government-required law.
FYI: the OP is in the US, not in Canada. Utah is no more free from this restriction than a rink in any of the other states. There is no law on the books against taking videos at public sessions, afaik.
Most skating programs and sessions in Canada are run by the skating organizations, which is why they comply with Skate Canada's restrictions. In the US, rinks run the majority of public and freestyle sessions, so the USFSA and ISI cannot impose those restrictions. Rinks also deal with USA Hockey and US Speedskating, who might have their own rules about videorecording. Most LTS programs today are also rink-run, which is why the ban on recording is being imposed by the rink. That's the key: parents of skaters complain to the rink management that they don't want their kids being recorded in the background. It's not that the club is asking for the ban on behalf of US Figure Skating; their own customers are making the request and as we all know, the customer is always right.
There were several parents in my oldest DD's class who refused to sign video/photo releases for sports teams because of their own reasons. (In one case, the husband was a high-ranking law enforcement officer and they were concerned about organized crime retaliation.)
There are some US clubs that still have the membership and assets to own their rink/rent ice time and run it as they see fit, but those are few and far between now. It was the model before the 1960's, but times have changed now.