Don't really agree. In a startup, CTO is often the most technical co-founder and may be a manager, may manage VP Eng, but the role is not well defined; it's whatever the technical co-founder feels most adept at.

Later on, the situation may be inverted, i.e. CTO might report to VP Eng. Usually, VP Eng reports to CEO. The point is that operational management of engineering usually flows from VP Eng, and not very often from the CTO. If the CTO does manage, it's a small shrub on the larger org tree.