I try to give 30 minutes to each direct report every day. This is not to "manage" them. This is not to "socialize" with them. I do not see them outside the office, and they are not my "friends".
Each person becomes invested in the job. They become better performers. They feel that the company is important. At 11, there are no longer enough hours -- so, if the number of direct reports gets to 11, you need another management layer.
The reason for this? If this is done, this is the easiest way to get maximum performance from the team members.
In your analysis, my team members each spend 30 minutes talking with me -- but 7.5 working. And, working more effectively. I am never surprised and schedules are always met.
What do we talk about? Process improvement. What is going wrong, with suggestions to fix it. Especially introverts, who can see what is happening, but are loath to communicate.
Using this simple approach can give a "10x" multiplier to a team. The "kinds of people that can get work done well, in a steady state" also benefit. By telling me how to improve the "steady state".
FredW