Your horse is adorable. I love saddlebreds. They tend to be very honest and try really hard. The kicking out to the side is something I feel like is most often tied to discomfort and/or frustration. Right now she is getting herself pretty stuck in a “frame” where the head and neck are set in a specific shape but she doesn’t know how to relax and swing through the back. That takes any horse time and even longer with a horse that has a long swan neck that makes it easy to curl. Judging a trainer on snippets of a single video feels unfair. What I will say is that saddlebreds often thrive when they are allowed complete freedom with their head and neck. Navigating how to ride her on a soft to loose rein walk/trot will teach you two as a pair how to maintain a steady rhythm and basic balance. If you take this approach and allow the head to climb as needed during the canter, I think you will have a horse that is much more willing to canter. If your trainer cannot see this and is not eager to hop on and show you how to trust her with her head and neck, I’d recommend softly looking around for other help. Many dressage trainers are not familiar with ASBs. A horse is a horse but it is helpful to have someone who has experience with the breed, arabians, or DHH and understand how easy it is for the long necked types to curl and get stuck. Months of regular work under the guidance of a trainer should have you much further along in terms of the horse understanding contact and how to stretch into the hand. This trainer may simply not have the skillset to help.
The fact that bucking has been established would also make me really really want to rule out pain first. Anyone who can look at a horse and say “it definitely isn’t pain” is absolutely blowing smoke. Horses can be fat and glossy with grade 4 bleeding stomach ulcer. Horses can be non-responsive to back palpations and have kissing spines. A horse can readily take a bit and have significant dental issues. Horses can stoically stand for mounting while the rider sits in a saddle with a broken or twisted tree.
Saddlebreds that end up in the saddleseat world are exposed to a set of circumstances that would blow the brain of many other breeds. Even if your girl never saw a saddleseat, she likely has that same good good brain bred into her. I’d believe her that something hurts before I took the word of someone who isn’t a vet say that it is the memory of pain. A lameness evaluation with flexions, gastric scope, and neurological exam at a university hospital or haul in clinic may run you $1k give or take. That’s not small money but she’s costing you board each month regardless of whether or not she is rideable and enjoyable. Sometimes it is better to get the answers all at once than dribble out money trying to take a cheaper route.