Some directors may have dozens, if not hundreds or even thousands, of people under them directly or indirectly. This rapidly grows beyond the scale where the director can jump in for anyone in the tree beneath them.

Your comment probably applies for a team lead with a half dozen IC reports. Add in one other layer (i.e., a half dozen team lead reports) and it might still apply in some cases. But double that number or add in another layer and things are probably far beyond the scale you were imagining. The manager’s job now requires a vastly different toolset that most developers know nothing about, have no experience with, and generally wouldn’t even know how to think about.