Generally those who have participated in, and are intimately aware of a given industry, are likely to be interested in staying involved with it, and often have the skills and expertise to succeed within the industry from another angle.
Top 26 Former NFL Players Turned Coaches - this list is interesting to look at.
There is a distinction between a general manager and a head coach, and what exactly that role looks like from organization to organization will vary.
You're in no position to say with certainty what the "absolute #1 skill" for a head coach in the NFL is, because that will vary depending on who you ask.
Your notion that teaching players techniques and skills is never the top priority trait in a coach seems slightly absurd. Again, this will depend on who is hiring and what they are looking for.
What do Belichick and Saban often preach? Things along the lines of, 'you've got to know your technique, you've got to be fundamentally sound, and you've got to do it the right way on a consistent basis'.
You'd be well-served to speak less generally and broadly, and give space for others to share their respective experiences rather than relying your own assumptions. You make broad, sweeping definitive/concrete statements all the time, which will never be true in every instance because the world and people are nuanced and complex.
Your perceptions, assumptions, and experiences are not indicative of others' experiences; your experiences are only a reflection of, well,
your experiences, which are totally legitimate and fine to share ... but have the awareness to realize that doesn't extend beyond yourself in every instance.