There are 3 group of horses in Hungary regarding their sizes. A pony is under 122 cm height. A small horse is between 122 and 148 cm. A „normal” horse is above 148 cm.
In my opinion a normal horse is between 122 and 148 cm, so the so-called small horses are the normal ones. It is the natural size of this animal, which is capable to not just survive but work without artificial feeding. A horse of this size can be very agile, swift and easy to control. From the aspect of war the most important attribute of a horse is endurance. Not speed and not weight for charging power.
I had the opportunity to compare the western European style of „high” riding on big (English thoroughbred offsprings) horses with the steppe style on small hucul, Arabian and konik horses.
My experience is that theese smaller, more archaic horses are better for mounted warfare than the big race horses.
However, if You don’t believe me, read the book „Small Horses in Warfare”, written by a British cavalry officer in 1900.
The worst thing You can do for practicing mounted warfare is to go to a „civilized” stable and start riding in saddle on a big, nervous horse while the coach repeats: Stay on the horse!
I started this way, but changed after two weeks.
I have found a suitable place, which was something like that: go to a horse keeper who has no stable and start riding bareback on a „small” horse, while the first instruction should be this: Fall off the horse!
You are right, it is quite difficult to find such a place.