Yeah, stretchies seem really weird at first, but really they are quite safe and acceptable in the Saddle Seat world.
Eliza looks great in that video! She's seems like a pretty level headed Saddlebred. I've never ridden saddle seat myself, but I have a lot of friends that do and the University I went to had a lot of saddle seat horses. Eliza has some great movement and you can tell that she is using her mind.
If you really wanted her back you probably could get her. Even if they are shady people, you know the horse so it's not like they can trick you there. As long as you are really careful and have the right paperwork you could do it. It's not like they could just leave the state overnight, especially if they are a big barn. :) Don't give up! If you want her back, GET HER!