Toe pushing is taboo, especially at the lowest levels of skating, except for a toe-push for edge changes such as on entries to jumps. Every power skating class emphasizes the use of inside edge pushes, not toe pushes. That includes the crossover underpush, which I think the OP struggles with, iirc.
Word of advice: do NOT use YouTube videos of elite skaters as your guide to test or competition standards. I've seen many USFSA and ISI judges' test sheets with comments about toe-pushing. I've heard judges mention it to coaches after competition events. The ISI and USFSA instructors' manuals both say that toe pushing isn't allowed and that the skater shouldn't pass the level if they can't prevent it from happening.
I've watched many adults and children at test sessions. The adults that struggle are the ones that lapse into ape-skating during the warmups, between elements/patterns, and when they are leaving the ice. It's an awful habit that brings down their performance as a whole. For some reason, the kids keep it in check from the second their feet touch the ice until they leave. Something to remember: you're always on display when you're on the ice.
When I see the elite skaters clomp onto the ice, toe push and skate hunched over like monkeys, even during warmups, it makes me shake my head. I guess they figure the judging doesn't start until the music starts. That's less likely to happen in local/smaller competitions and test sessions.
I don't *think* they're doing it intentionally - could they be using it as a way to warmup their feet, by rolling waaaaaay up to the toepicks?