Race horses are sometimes a little more supple to the left but not always.
Race horses gallop and run approximately 1/2 of the time in the right lead -- always.
It is a totally false concept that they only go in their left lead. They are taught to change to the right lead (if they do not do it on their own) at the head of the stretch and the head of the back-stretch. If a horse does not change leads, it places so much stress on his left front leg that he will break down. When race horses fail to change to the left lead going into the turn or to the right lead coming out of a turn, the jockey or exercise rider will either pull him up or will at least tell the trainer when he gets back because it is frequently the first indication that a horse is sore in the leg on the lead he does not change to.
I used to retrain a lot of OTTBs and OTTQHs and never had lead problems if they were sound. I started out by doing a lot of suppleing exercises.
The first exercises I did were 'leg yielding' exercises. If you have not done these before, you need to have 'eyes on the ground' to tell you if you are getting bend throughout the horse's entire body and not just a bent neck while the horse just move over. The horse has to bend through his whole body. If you do this in a ring, try to keep the horse parallel to the rail as you 'push' him over. If I was riding in a ring, I did 'half circle and leg yield to two track back to the rail' exercises. You always want to leg yield away from the gate or barn or other horses. [This, by the way, is the first thing I teach a green colt -- as soon as I get good forward movement -- like the 3rd or 4th ride.]
It is very important to supple ex-race horses because they are really bad about dropping a right shoulder when going to the right. They are very 'good' at making a rider use the left rein to keep them from 'falling into' a right circle or using the right rein when going to the left.
If your inside leg cannot keep your horse from falling into a circle, he is not 'centered' enough to expect correct lead departures.
The opposite is also true. If he is stiff and you have to pull on the right rein to get a right circle, he is still not centered. If you have to 'pull' a horse into a circle, he will almost always take the outside lead because when he pulls on you, he is putting his weight into his outside shoulder. This is also true on the longe line. Any horse that is pulling hard on the line will almost always take the outside lead because that is where his weight and energy is going.
If you get this horse to leg yield both direction and he stays centered but still does not pick up one lead, then you need more hip control.
One way or the other, they ALWAYS TELL YOU when they are ready to pick up leads correctly. When you ask them prematurely, you risk them making a 'thing' about a lead and making up their minds that they WON'T take a certain lead. Then, you have to do a LOT more suppleing, obedience and centering exercises like you have to do on any other horse that is spoiled on leads.
Ps. I though I would also mention that when re-training horses off the track, I always taught and used a 'one rein stop' a lot. I used one rein instead of both until they were ready to stop by barely picking up both reins. I NEVER tried to force or 'pull' a horse into a stop with both reins until he was willingly stopping. When I teach a one rein stop, I DO NOT let a horse disengage its hind end as part of a stop. I do not want a horse 'throwing' its butt out. I just it them to 'give me its heads' and STOP.
Hope this helps.