But you don't now need to introduce additional abstractions when you go from 50 to 125. Why introduce a VP of engineering? Can the CTO not handle 3 extra director reports (4 if you add a "director" of QA with 2 reports?) Come on. Who is going to come on to be the "VP of engineering" for a startup, yet not have/want oversight for Security, Analytics, and Infrastructure?
Also in the 50 person startup, the product lead with 5 reports is a "Director" but a product lead with 11 reports is a "VP"? Get outta here, this is silly. It's the exact same role. It is NOT more scope or oversight.
There should be orders of magnitude size growths before you start overthinking the difference between a director and a VP, or introducing new layers.