Sorry, I started the previous reply before DrafttyAiresMum posted. >< TBH she is actually very smooth to ride. She looks very choppy in the video because when she's lazy-footed like this she tends to just jam her feet into the ground instead of stepping down lightly. She is on an EPSM diet now and has improved somewhat in the stabby trot department.
Also, if you look closely at the video, you will notice some off-tracking in the right hind leg. She was a rescue and was found with a large, untreated puncture wound in her right hip. It looks like she as two points of hip on the right side. One lump is her actual point of hip and the other is a giant lump of scar tissue. She has been examined by multiple veterinarians and all agree that she is sound and that it should improve with further conditioning.