Cavalry horses were trained en mass - hardbroke, but not even greenbroke by our standards. As a result, the horses were unruly, unpredicatable, and were why the Indians ran circles around us.
That and movies is where we get the "cowboy broke" mentality. In reality, horses were only broke that way for a very short period of time - but it has been dramatized in the movies, so when we think of breaking horses we think of bucking broncos, but that period of time covered maybe 50 years at most - we have had horses for thousands of years.
The commercial trainers of today have taken advantage of our warped vision of what horse training used to be in the "old days". They have generated the foolish term "natural horsemanship", which is the way horses have been trained for thousands of years other than the 50 year "cowboy" period. In addition, they have warped the whole "natural horsemanship" to encompass what I term "quick fix" training - gimmicks and shortcuts that produce results, but do not modify and condition the horse's behavior. As a result, they are temporary fixes...without modifying natural behaviors you cannot expect those behaviors to not surface in the future.
Natural horsemanship is really nothing more than being able to truly (as in truly) reading your horse's behavior, having a deep understanding of horse behavior, and develop techniques to modify the behavior. Quick fixes are not permanent, discipline is not permanent, and whispering in its ear accomplishes nothing other than getting horse hair in your nose.
Years ago, people had far more horse sense than today. Because horses were used daily for transportation, agriculture, and other work, people were much more familiar with horses. They knew more about conformation - certainly would never assess conformation without putting their hands on a horse, they knew more about horse instincts and behaviors, and they knew more how to modify those behaviors.
Today, horses are, for the most part, part time hobbies. People are specialized - you have riders, trainers, vets, chiros, and so on and so on. The need to really be an equine expert just isn't there any more. While that is in some ways good, it means that the vast majority of people today just don't have very good horse sense. Oh, they can spout "give to pressure", and they know what it means when a horse swishes its tail or lays its ears back, but very few people really know any more than the very basic horse instincts, why they behave as they do, and how to modify those behaviors. You have to know the underlying reason for a behavior and be able to modify the reason for the behavior - not what is done today which is to only modify the behavior itself. Without modifying or resolving the underlying reason for the behavior you are doing nothing more than giving an aspirin for a headache...you are not resolving what caused the headache to begin with, and as a result the headache will return at some point in the future.
I'm not being critical of anything or anybody. I am just answering the question posed by the OP as to "old time" horse training. What most people think of as old time training is actually not true. Again, our perception is of the cowboy days, but that was so short a period of time it is actually inconsequential...it has just been glamorized.
Probably too much elaboration...