I don't think you can tell enough about her back from the "conformation" picture you have.
What I saw in the video was a tense, stiff horse with a pretty decent range of movement for being as tense as she was. The rider appeared to be "setting" the head rather than teaching the horse to move toward the bit and over her back. However, if you look at the trot and canter work, the horse is still flexing at the sacroiliac joint (not always, but when she relaxes a bit she does), stepping well under itself, and has good articulation in the hocks. Has decent jump in the canter for what's being asked of her. Good basic gaits from what I could see. Walk seemed a little lateral. That could be due to tension. I'd want to see her walk out in the pasture or something to determine if it's the result of the rider or natural.
Rider seemed to be a hand rider mainly, and causing a lot of the tension in the horse. Hard to say for sure though. It could be the horse was very tense to begin with, and rider was reacting to that.
I don't see Fourth being out of the realm of possibility for this mare. What I see as the biggest issues is she doesn't have any basics and she's 10. You'd have to start from scratch with the mare.
Also, unless you're an Arab fan, don't buy an Arab cross. She might be good-minded, but that doesn't mean she isn't thinking about 20 ways to evading working correctly. You won't get to repeat (in a row) an exercise with an Arab cross. Warmbloods you can usually get away with doing an exercise a few times to get it right. Arab will have you at her mercy if you try something more then twice. You really have to be a smart rider to stay two steps ahead of an Arab cross. The people I've know who owned Arab crosses either love them or HATE them with a passion.
I like the mare. If the price were right, I'd buy her (with the intention of taking her though Fourth). However, her lack of correct training in combination with her age would make me shy away from buying her.
This is just my opinion.