Secretaries only help create value insofar as they genuinely help take administrative workload off the shoulders of people whose skills are better put to use elsewhere. In the real world, you need engineers to communicate their own work, because there's a strong overlap between "people who can read and understand code" and "people who will earn more money with an engineering salary instead of a secretarial salary". By hiring someone else to push paper, you actually impede communication rather than improve it, because now there's an unnecessary stakeholder in the middle.
I actually think this other comment on this thread says it best: https://news.ycombinator.com/item?id=36326151. I've worked with a bunch of "Tom from Office Space" product managers in my career.
I've seen lots of product managers who basically just "facilitate" between stakeholders (e.g. between engineering, business execs, and customers) but don't actually add much value to the primary job of product management, which should be defining, in detail, how the product should actually work.