No not all leaders in my experience are calm and level headed. The best/most effective leaders use what style works best for them. Coaches and team captains as you state are no exception. Screamers can be positive leaders and be loved by their charges. Calm and quiet people may be that way for a reason. Maybe they don't know anything and don't want anybody to know. Leadership by example is not just a slogan. I describe my leadership style as "follow me." I may push but I'll be doing it from the front.
While still on active duty I loathed being late. Still do. I couldn't imagine having a uniform item out of place. I never needed a haircut or a shave. If I didn't know something, I learned it so next time I knew the same as everybody else in the room.
It's hard to have people "follow me" if I don't present myself as somebody worthy of being followed.
In my opinion, a big part of leadership is just being there. What I mean is that you need to be out and about looking at what your people are experiencing and/or trying to overcome and letting them tell you what is going on. This is priceless and they will remember. this might not be the case on a HS sports team but is more important in the fleet and in industry.
Leaders show the way! As a CO, I had a standing offer to my people that I'd buy a beer for anyone who could beat me on the PRT. No, I wasn't a physical marvel who spent half the day in the gym and actually I was old (40's) and could have stood to lose some weight but I did prepare for the PRT and I expected that my people should do the same. I was pretty good on Situps and Pushups which really helped my scores but as a LEADER, I wanted to demonstrate that I wasn't going to expect more out of them than I'd be willing to do myself. Now, as a college professor, when I give my students a project to complete, the first thing I do before they even start doing it is demonstrate how I would/did do it. For me, it does head off a lot of questions and for them, it gives at least one method of handling the work.
Leaders CARE! You might really need every hour of your people's time but when someone's child is sick or Seaman Shmuckatelli's grandma who raised him has just died, a leader needs to allow them to step away from the mission as needed (and physically possible). People need to eat, sleep and care for their families. I'm not saying that I won't work people very hard but sometimes you need to make exceptions. By the way, when your people are working extended hours, YOU should be visible and helping/promoting/doing what needs to be done to ease their efforts even if YOU are not always needed.
As an example, one time at sea the ship was scheduled for a missile exercise and the night before the exercise, the launcher had
mechanical problems. In the wee hours of the morning, the Gunners Mates diagnosed the problem and needed a part from the
ship's storerooms. They woke up the Duty Supply person who refused them and told them to come back during working hours
which would mean that we would not be ready for the exercise. I (LTjg) went to the Supply Officer (LCDR) and woke him up to get
him to intercede. He was moderately irritated with me and more upset with his people. His Senior Chief was more irritated with me
and even more angry with his Duty Supply. The Captain was happy that we were up and ready when the missile exercise took place
and my people were proud that it all worked and thankful that I had their backs.