Restarting OTTBs is what I do, and while most do have a nice trot under saddle right away, some can be more tricky. I had one stunning gelding that had all the right conformation to have a beautiful trot, but would only do a tiny “pony trot” under saddle. I worked with a dressage trainer w TB experience a few times, and she had me trot in a circle that had two or three random poles placed around it. She asked me to get to the pole “wrong” like the wrong distance to a jump (too short, too long, etc) and after getting to the pole “too long” several times, the horse learned to extend his leg. We worked on that a bit and then I was able to translate that to working around the ring without poles. He ended up having a stunning trot with a great extension.
I also use several trot poles in a row in a fan shape on a circle, to teach them to collect and extend (Jimmy has these exercises in his gymnastics book I believe)
Another mare I had, I actually posted on here about and got some great ideas. She would also pony trot despite having a beautiful trot out in the field. Difference with her is she just wanted to Canter all the time, and would do a pony trot/Canter trantor. So I resolved to only Trot, and do tons of suppling exercises, to get her to bend her rib cage. So, serpentines, shoulder in, leg yield, tiny circles and figure 8’s. She was not allowed to canter. I kept my leg on, and kept the posting rhythm that I wanted, not what she was giving me. It took about 20 minutes of her having a hissy fit, then she yielded and gave me the most round, lovely, supple big trot!
So, to sum up, keep your leg quietly on, no matter what the horse does. Don’t let them fool you into taking it off! Suppling exercises to get bend through the body, and some randomly placed poles may be a big help. I also like to work my horses in a round pen with either a neck stretcher or Pessoa rig to encourage them to use their backs before riding.
Good luck! The Trot is in there, you just have to find it 