Trainers that 'push' horses are usually competitive 'event' trainers. They typically put a lot more 'pressure' on every horse they train because they have a goal and a specific job that their horses need to do well. This does not mean they are abusive or 'rough'. It means they have to expect more from every horse they ride. Many are riding toward age related goals -- like all of the trainers that have owners nominating these horses for futurities and derbies.
This system of age related events causes many trainers to put more pressure on some horses than their bodies and/or minds are ready to handle. Getting a horse to these events costs a lot of money in both nomination fees and training fees and since many of the prospects are bought at 6 figure prices, it is usually the horse owners that keep putting pressure on the trainers to get more done faster.
This is one of the main reasons that I got out of show training. A trainer has to be able to 'read' horses very well to know haw hard they can push that horse. They also need to be soundness 'experts' because it is often the body that 'breaks'. But, they also need to be able to 'handle' owners. Their ability to make a living depends as much on handling owners as it does handing the horses.
Most recreational riders never get close to putting enough pressure on a horse to 'blow it up'. As a matter of fact, they seldom put enough pressure on their horses to get them doing very much a even a moderate level, much less a competitive level.
The 'key' to training horses is to make every ride count. You should strive to have a horse doing something better at the end of every ride than it could do at the end of the last ride. Knowing when to 'quit while your ahead' is one of the greatest tools any trainer can use. Never ask a horse to do anything that it is not ready and able to do. Then, once you ask, make sure that the horse complies fully.