1 VP per
8 directors per
64 team managers.
So, how are there more openings at higher levels?
1) in those 8 person teams you mention there may not always be a place for an expert IC
2) companies often reserve the IC-expert budget slots for internal IC promotions while relying on management slots on external recruitment to a higher degree
Additionally, it can be easier to transition to an equivalent or higher role from one company to another on the management side. Most companies are open to hiring a VP from outside but often the highest levels of IC roles are only filled by those with significant experience within the company.
The knowledge and ability to do tech work is great, but many jobs require that AND some other skill set (sales, research, communication, people management, etc). Those jobs also tend to be very well paid.