IMHO, to try to rein on a horse that is not bred for reining (or at least for cutting) would be like trying to make a stock dog out of a beagle. You can probably get a beagle to do a little of what you want, but a dog bred for many generations specifically to do a certain job well is completely different than trying to invent it over with a dog like a beagle. Same could be said for trying to hunt rabbits with a Pomeranian.
You cannot live long enough to breed the more talented individuals to the more talented individuals and come up with anything even close to the least talented reining bred horse out there.
It takes several things to make a good reining horse. Of course, these things are usually present in a well-bred prospect. You need strong, low-set hocks, a strong loin, short cannons (low set knees) and a smooth pleasant way of moving with its shoulders up and its head down. A reining horse needs to be a superb athlete that is very quick-footed and very coordinated. Probably the most important things a good reining prospect needs are a VERY good mind that lets you put a LOT of pressure on him and a very good mouth. Reining horses need to be 'micro-managed' in every little way. They have to completely give you every tiny part of their body and their mind. Even to rein at a low level requires all of the above.