Yes that’s how analogies work, they’re not exact equivalents.

The point was to illustrate that the actual value creating job of every employee from the bottom up to the CEO is fundamentally delivering products or services. An organization that takes its eye off that ball and hires people specialized in “people management” is well on its way to the remarkably common decline and fall of large corporations. If you hire people whose specialty is interpersonal drama then guess what you’re liable to get.