Well, remember a while back that I posted the pony has a mental block about trotting on lead? I've never worked with her on it.
I would suspect yours is just being lazy, she will do it, just reluctantly. There's no ~reason~ someone would have taught her to walk slow on lead. If you want her to be more responsive to trot, use your body language, voice and lead line, say TROT, kiss to her, or whatever you do and then use a lunge whip or a shorter stock whip, behind you like the trainer did to smack her as sharply as needed to get her to trot... without you having to run on ahead and drag her.
Trotting back and forth in a line is useful for her to know, it helps a vet evaluate her for lameness in an exam...
But I wouldn't view it as a necessity.... if the situation were ~really~ urgent, like a fire or a tiger comin' after her, I suspect she'd move faster than you want her to ;~D
Yeah you are probably right:') It just surprised me. It's no fun pulling a horse but I was in a position where to to back down would have let her get away with it.