Leadership is always an interesting topic. In my experience most head coaches come with the work ethic but I can't say that for all. The other three traits in my experience that matter for HCs are respect given to the player, holding players accountable, and then do the players trust the HC to put them in positions to succeed. How that translates to "leadership" is debatable.
In the NFL, I do believe the attitude of the best players on your team does matter. Are they about the scheme or themselves? Are they about putting in the extra work needed, physical and mental? Do they hold themselves and others accountable? When it comes to Larry and Peterson in these areas and compared to Watt, Butler and others, you can see a difference. In style if nothing else.
Hard to argue this, especially in sports leading up to and including high school. A little trickier at the collegiate level. A good college coach would be much better at noticing a leader, but if the 3rd string QB is the best leader on the team, he's definitely NOT being named captain. I've never played professional sports, but can't imagine it being much different than my college analogy.