The OP knows her horse, and if she feels that all he needs is some room to run and burn off some energy, I believe her. OTTBs are like that sometimes, as my video shows......they need room to expend that explosive energy.
Our boy is an absolute angel on the ground, obedient and respectful....same under saddle, but is very energetic and wants to GO...definately not a beginner horse...but still, he is a great ride and a lovely well behaved boy on the ground.
He just needs some room to let loose once in a while. So, I think the OP hit the nail on the head with her read on the situation. She has a young OTTB that just needs to be able to do what his genetic wiring tells him to do....run. Needing to do so does not mean they are badly trained, just energetic ex athletes.