Patience: it's an animal, it doesn't care what you want, but with work and education, you both transform.
Perseverance: incremental work, with a living being that isn't subject to reasoning, is what makes the most change.
Ownership and Obligation: the animal is your responsibility, it does not have responsibility, you own your failures.
Decisiveness: related to ownership.
Benevolence: xenophon's "nothing forced is beautiful," and (I think la Guerieniere's) "ask much, be content with little, reward often." exemplify these.
Takt: the ability to express your intent simply, unambiguously, consistently, and with a stable emotional presence.
Temperence, moderation, and sympathy: this is a living thing you can't just threaten into submission.
Magnanimity: the animal is going to test you constantly, and you have to be greater than it.
There are other aspects that the words we use today have been freighted with inaccurate critical interpretations, but they are similar to the Aristotelian virtues.
These (among many) in addition to exercising the physical experience of leading, which means to be followed willingly. You learn how you don't just read about something and become it.
The qualities of good horsemanship are essentially moral and physically cultivated qualities, and among those is the ability to be recognized as a leader and followed willingly by another creature in nature.
I am saying the qualities that people recognize as leadership and follow willingly, are moral qualities that can be developed and taught, and there is a history of those being taught through horsemanship.