I'd be willing to bet those boards also had a "secretary" and a "vp/co-chair" and a "treasurer" and possibly other positions, because "the one person who runs everything in the organization" is both too much for one person to handle, and doesn't allow people to specialize as well.

You have described a "leader" as a Social Connector, a Conflict Resolver, a Parliamentarian, and a Generalist. Why should these all be one person, instead of separate people, so they can all do a better job of their thing?

they think that their job is to just be assertive and motivate the other members to do their tasks, which is what the media often describes as ‘good leadership’.

Sounds like there are many divergent and incompatible definitions of "leader" and "leadership".

Thinking about the Cabinet of the executive branch also informs my thinking. Joe Biden is a perfect example of how all the departments would run just fine without him. Yet he is the foremost leader in the land.