The Leadership Dilemma

Why does anyone take on a management or leadership position?  Certainly some people like the power, some people like the relational elements of it, some people like mentoring, some people are administratively gifted and like planning, budgeting, etc.  That said none of these reasons alone are sufficient to hire another person.  You wouldn't spend your own money to employ someone for one of these reasons.  Ultimately I believe that people manage or lead other people so that they can multiply their effort.   A person may be phenomenally good at what they do, they may push the limits of reasonability in their work ethic, and they may have boundless ideas but there exists a limit to what they can accomplish on their own.  In order to accomplish more they have to manage or lead other people (or get happy not accomplishing but only a part of what they'd like to.... or get someone else to champion their ideas, etc.)

I've been thinking about this in terms of a mathematical function:

I'm sure volumes and volumes have been written about how to optimize first summation.  That is, how do I efficiently manage a team, what's the ideal team size and structure, how do I identify and develop the "beta" of my people, etc.  I'm interested in the second summation of the function with the "alpha" terms.

That represents the leftover time that a person has after pouring themselves into leadership/management.  In other words if a person would passionately do tasks X, Y, and Z on their own is there any time left to do some of those things after they're done leading.  Mathematically if 100% of a leader's time is spent managing then 1- the summation will be 0 (i.e. no left over time).  On the other hand if 10% of a leaders time across all of their people is spent managing them the manager will still get to spend 90% of their time doing things that (presumably) they enjoyed doing as an individual contributor.

I want to be a participatory leader.  I want to have time left.  I don't want the cost of my alpha to be 1 leaving me no time/energy to do anything else.  I like the creative/building process and leadership isn't worth it to me if I don't get to participate.