Any breed of horse, any size and any color can be taught to do anything really. It all just a matter of finding the right way to do it for the horse. With laying down I don't really believe you need to have a good reason for it. The horse doesn't have to be dangerous to need to lay down. I love to challenge my horse and work on anything I can think of, and not everything is from the saddle. Teaching a horse to lay down and be comfortable laying down with you around is something I find to be very helpful and is an added safety precaution. My horse knows how to lay down, though I didn't teach him the way above, it took me well over a year to teach it to him but now I thank god that I did. Over the past five years I have had a few accidents in which Jake fell (once over jump, got caught in barbwire, tripped and fell) in which he laid down and stayed perfectly still until I was able to crawl away from him, get him untangled and check him over to make sure he was okay and waited until I asked him to get up. It has saved both of us a lot of possible situations where he could have hurt me in struggling.
Back to OP, I have found with teaching the lay down there are over a hundred different ways to do it and there is one way to do it to teach to your horse (safely) as each horse is different. If your going to do it be patient and Take your time, do not force it.