Every other source I've come across says the rinks pay a blanket fee, the network pays a fee, but nothing about the skaters being held directly responsible. I'd like to see what sources led you to the conclusion that the fault goes to the skaters.
Because the ISU makes the skaters sign something that says they have the rights to skate to the music. K/F must have signed it and, in their case, it wasn't true.
Sure, the networks might CYA and do extra work to make sure they have the rights to play music by very famous artists. But are they going to check each and every skater's music for every event they broadcast? Enough of the time the skater doesn't even identify every single cut or identify the cuts they do mention with enough detail or accuracy for the network to do that anyway.
Now is this the way it should be? That's a different question. I think the skating community doesn't think about these things much because:
(a) they don't know much about copyright issues (and what they do know is probably at least partially wrong)
(b) for the vast majority of skaters, their programs aren't going to be broadcast to a wide audience so it doesn't matter
(c) 99.9% of the time, the rights to their music were covered by blanket fees that rinks, arenas, and (for the top skaters) the broadcasters paid
But these days, when artists aren't so dependent on record companies and are experimenting with new ways to distribute their music and with streaming services showing every single skater in the event, and fans putting their own clips up on public websites like YouTube, I think there are going to be more cases where the blanket fees don't cover everything and the artist finds out about it.
So they need to think about it now.
I viewed that sentence as an opinion, rather than a statement of actual fact and culpability. It demonstrates one interpretation of the law, while others, as
@tony has shown, have a different interpretation of the law and whether or not the skaters themselves bear any responsibility.
When it comes to the law, it's all opinion. The law has a lot of gray areas.
To focus on K&F is kind of putting on blinders and refusing to see the larger issues, which is what we need to see to understand why we are getting such poor figure skating coverage. K&F can't fix the situation by any action they can take. And fixing it is what we need. NBC/Peacock is declining to host the replays and USFSA is not doing enough to get them to change that. To me, those are the issues, as well as the larger one that there just isn't enough of an audience for the replays to get NBC/Peacock's attention.
I see these as two different issues. Even if K/F's team blew off the group when first contacted and then K/F hadn't skated to that music at Worlds knowing full well they didn't have the music rights, skating coverage for the causal viewer and for those who don't have cable and/or Peacock still sucks.
It sucks even more now without replays though and I see that as on K/F's team for blowing off the issue when first contacted which led to the lawsuit and for K/F for skating to the music at Worlds when they knew full well they didn't have the rights to it. I don't blame them for skating to the music before they knew. I blame them for skating to it at Worlds when they knew. I see them as forcing the issue and causing NBC to panic and do the most risk-averse thing -- which I also blame NBC for as the reaction was OTT IMO but the catalyst was K/F
They get copyright-claimed and taken down. A lot of this stuff isn't all that complicated. 
