When I am working with any species of animal, I will work to balance instincts and responses. Reinforcing innate drives and motivations will make them stronger. This is not always a good thing. It is, if you are working them and need to develop that drive, but if you think about it, you develop a specific aspect and application of it, you don't just reinforce it as a whole, or the animal is likely to perform it whenever he wants to, regardless of whether it is the right application or not. You want to reinforce the motivation when in the correct context, not generally. It's about putting it on cue so that you don't get it when it's inappropriate. That cue may be anything, it could be a voice cue or visual cue from you, or it could be cued to an environment, particular place, or specific circumstances.
A racehorse will race when he is on the racetrack. A sight hound will chase when he sees small animals moving. If that drive is reinforced without any focus, how difficult is it going to be to interrupt it? Very difficult. The behaviour is self rewarding, and the more it is practised, the stronger it becomes. If that behaviour is tempered with training so the horse or dog can interrupt it, is taught where to perform it and where it is not safe to do so, then you can balance those motivations. If left unchecked, the behaviour will not be a balanced part of the personality, but an overriding trait that will be difficult to counter. It is not much fun if your racehorse thinks that a canter always turns into an all out race, and every long stretch is for running as fast as possible. That does not make for a safe or enjoyable experience. Neither will you enjoy walking your dog if he chases off after everything in sight, and you have to continually go and find him.
Teaching impulse control is a very important part of learning, and invaluable for being safe around any animal. This means that the racehorse can interrupt his instinct to run, and the sight hound can interrupt his instinct to chase, making both animals safer and more reliable. It also means that humans have the ability to keep him, and those around him, safe. The only way to achieve this level of safety, and control of the situation and the animal's behaviour is if the horse or dog does it himself. You are not, and never will be in control of his mind.
There is no denying it, if your horse has taken it upon himself to race, you are going to have a hard time trying to stop him on your own. And by on your own, I mean without any input from the horse. Yes, you could use all your aids with increasing force, try to turn him so that he has to stop or run into something, but this should be an emergency situation, not a normal part of your interactions. It is not safe, has a created negative associations for the horse and the behaviour is not likely to change if it is regularly repeated. Besides, any horse is far stronger than the strongest human, so there is every chance that he will be sufficiently motivated to ignore whatever force you use to try to stop him. What options do you have if you are on a horse that is immune to all you do? One. And that is to teach him to manage his own behaviour, have awareness, and learn lots of exercises and routines that you can then employ to diffuse the situation safely.
The same applies to your sight hound. Off lead, this is imperative, as there is no chance of running after him and regaining control of the situation, he'll be gone, and your two human legs will not keep up! On lead, it may seem as though you have control, but how many people have their arm pulled off every time their dog sees something and decides to chase? His chase is cut short, but only at the expense of your arm and his neck if wearing a collar, or body if on a harness.
Control is something you perceive you have over animals, but in reality, you do not. You cannot control their thoughts, instincts, motivations, or actions. You can shape, balance and guide these things, and the best way to do this is through positive reinforcement. The result is that they will find what you teach them enjoyable, which means they will be happy, even eager to repeat these behaviours. And that is what will give you the confidence to know you can successfully and reliably manage your horse or dog, whatever situation you both find yourself in.