Many benefits abound from both doing and teaching Karate. I started teaching as an Ikkyu in our own dojo and a Shodan at another. I learned about:
My Strengths & Weaknesses
The art of communicating in different terms to different people
Did I mention Patience?
Working with Adversity
Taking it to the limit one more time...
Etc, Etc, Etc
Because my son grew so fast, I first placed him in gymnastics to help him become more comfortable with his body (he learned a lot about centering); then, when he was too big for gymnastics (and not exactly olympic material) at 8 years old, he started taking martial arts.
The change was amazing in HIS patience and attention/focus. He began to really excel at school and became more outgoing as he gained confidence. My favorite memory is watching him as an 11 year old brown belt teaching a 60 year old white belt! I think that speaks volumes.
As to the spiritual side, perhaps many do not talk about it because it is not a spiritual "way" for them, but more as a physical training way. Sensei Thompson remarked once there are those who work out and those who train. We train....