in Emergency Backup Mode
It doesn't need to be live-streamed. You can start the video with a shot of the skater holding up a phone/labtop that displays the official world clock time (maybe even through a specific link on the ISU website for further foolproofing), and then keep that phone/labtop in frame of the camera to verify the skater's performance is taking place at the set time they have chosen to compete.
Oh, because there's no way that could be manipulated at all.
Again, you give each a competitor a set time that works for them, where they must compete and verifiably record their performance. After all of the competitors have done this and the performances have been sent to the ISU, then a time is set for judges to do their work.
Except I'm yet to hear of a secure, trustworthy way to "verify" that the skater a) only recorded once and b) skated at the time required, without live streaming.
Judges are normally traveling outside of their timezone and have to be "up at 3am" in another location from what they are used to, so it shouldn't be too hard for judges to arrange their schedule to judge an event from the comfort of their own home. Possibly the judging times could be staggered too though, it's not like their marks would be public until all of the scores are set, the judges themselves could be known only by the ISU as well, until the date of broadcast.
Judges travel to the timezone a few days in advance to adjust to the timezone (and attend practices). Judges are also volunteers and many have full-time jobs. Leave could be hard to come by for a lot of them.