Generally, I'd expect the CTO/VP/Director to have more management experience, but in a lot of these cases, they may have the resume, but not the skills.
Has anyone had a career from selling management coaching or consulting on these issues?
In a startup, you don't even need to get promoted into management. A founder who has hardly any experience as an employee, let alone as a manager, can just declare themselves to be the CEO or CTO. I suspect that such lack of leadership experience among founders contributes to the failure of many startups, once they get to the point of scaling up and hiring employees.