I may be nitpicking, but ... Coaching and recruiting are separate but both the head coach's job and responsibility, and on his job description. That doesn't make them the same thing, and the coach wears the different hats at different times. A coach does not recruit during games or practices, and he does not coach when recruiting young men.
A coach does have several individual jobs as part of his job as "Coach", as you mention, and coaching and recruiting are just two of them. I believe they are distinctly separate, and recruiting is more important, certainly more than checking a box.
"Coach" is both a noun and a verb, TMT. What does a coach do? He coaches. If recruiting was separate from coaching, he'd have the title "Head Coach / Head Recruiter" on his office door. He doesn't, because recruiting is part of coaching.
Everything that he does within his capacity as the coach -- recruiting, game prep, running practices, administration, game management -- falls under the rubric of coaching. Saying that he isn't coaching when he's out recruiting is like saying that a teacher isn't teaching when he/she is grading papers at home, or that a cook isn't cooking when he/she is chopping up the vegetables. Or, to use another analogy, saying that a coach isn't coaching when the public can't see what he's doing -- since it appears that you guys seem to equate "coaching" with game management -- is like saying that an iceberg only consists of the one-tenth that you can see, while the other nine-tenths of the floating formation of frozen water is something other than an iceberg.
Coaching and recruiting are not separate. Recruiting is a part of coaching. To say that one part of what a coach does for a living is "coaching" and another part isn't is not only arbitrary, it makes no sense at all.
Coaches have to recruit to be successful. Coaches have to teach to be successful. Recruiting isn't coaching, but coaching involves recruiting. You're both right. Let's move on.