The short answer to your question is that a HC has to be as much an administrator and manager as they do a coach. They vary a lot in how much they manage versus coach, though. Ideally, they're focused primarily on coaching the coaches, making sure they have the tools to do their jobs effectively, and making final decisions on rosters and schemes. Many also take playcalling duties even if they're not the primary designers of the offense and defense.
It's a complex job, which is why it's rarely a good idea for them to also be the GM.