Honestly, you don't know what you are talking about when it comes to dressage.
The fact that you even think it is about some kind of "frame" or riding style says it all...
I don't think it is about a frame. I don't like using the term "frame" with ANY riding. However, if they have been "pushing a little to hard", then I wouldn't be surprised if they have been trying to put her 'in a frame'. And if I tried that with any of my horses, I'd expect "crow hopping, incessant backing, refusing to be caught, cinchy".
"The beginning of dressage IS precisely how to teach a horse to balance itself."
The beginning of ANY riding is teaching the horse to balance with a rider. But if the horse is "crow hopping, incessant backing, refusing to be caught, cinchy", then MAYBE they aren't really following "Pohdjasky, Steinbrecht, and Seunig".
This is not an argument about dressage. It is trying to get a horse off to a good start - and I frankly think that means dropping WP, dressage, hunter...or in terms of the OP's interests on her page, "Bareback, Classical dressage, Companion horse, Dressage, English pleasure, Equitation, Halter, Pleasure trail riding, Showmanship, Western pleasure".
I have little tolerance for the English vs Western garbage arguments. At the Riding 101 stage, there isn't much difference between any of them. They are ALL too advanced for the horse if the horse has balance issues.
And you can allow a horse to learn her balance while riding in any saddle, bit or bitless, etc.
"Maybe it took you that long. That doesn't make it a rule."
I haven't met anyone that is as secure in an English saddle when starting as in an Australian or Western saddle. After all, they were specifically designed to help a rider stay on a half-broke & rebellious horse.
Maybe I am a slow learner. And maybe I like keeping my shoulders intact. All I know about the OP's riding ability is that the trainer doesn't want her to ride English on this horse because "my trainer doesn't want me falling off if she does something silly".
That tells me the OP doesn't have an incredibly balanced seat and that some help from saddle design is appropriate - according to her trainer, who has seen her ride.
I bought an Aussie-style saddle when I hurt my back a few months after taking up riding. For the next 6 months, I couldn't lie down on a bed without stabbing pain. Mounting a horse was a huge challenge, and dismounting sometimes ended with my clinging to the horse to stay upright. I probably shouldn't have ridden at all during those months - but I did.
And with the help of an Aussie-style saddle, I stayed on during a few bolts, and multiple leaps & spins & whirls. It is no more wrong to use a saddle design that helps you than it is to wear a helmet just in case you fall. Both are equipment to improve safety.
And if I close my eyes, I cannot tell the difference in seat between my Bates AP & my DownUnder Master Campdraft. But when the horse hits the fan, I sure can tell the difference!
I'm not attacking dressage or WP, or ANY riding discipline. I am suggesting the OP focus on letting the horse learn her balance in a relaxed, no pressure setting. There IS a difference in how English and Western styles approach using reins, and how the saddle feels to the horse, so changing tack on a horse that has balance issues isn't helping. Pick one and stick to it. When the horse is balanced, THEN you can decide what sport to participate in - if any.