It hasn't yet come to me where you are coming from with your discussions that highlight the concept of zero. But maybe this time will be different. An answer to your request for a description of what exists at the center of gravity seems quite straight forward, though a little tricky for a layman to put into words that PhysBang will let pass. Would I be missing the point if I use a non-space time description to say what exists at the center of gravity is a location in space that represents the point where all of the net gravitational "downward" forces of all of the components exerting gravity within the volume of space under consideration would net out to zero at any given instant in time?

Further, the pencil analogy simply means to me that in any gravitational environment, there is a constant that can be added to the equations to make the pencil stand on end, but that is only a mathematical constant that would achieve the balance given the exact gravitational environment, the arrangement of matter, at an exact point in time; the constant would represent a mass necessary and the location of said mass that would solve the "balance" equation. A new math "fix" would be required for each instant that you want the pencil to stand on end, and that would require a new mass and location for that mass every instant as well. Such a constant is a "fix" to the reality that no gravitational environment can exist that allows a pencil to stand on end for more than an instant. The "why that is", again from the perspective of a simple layman" (I'm always tempted to say "simple pie man, for some reason, lol) (leave me along PhysBang) is because the center of gravity is not a fixed point in space, but a point that is continually being refreshed as a result of the relative motion of the objects in the given volume of space. For practical purposes, they never arrange themselves in a way that will establish an enduring fixed center point.

The problem is similar to the vacuum energy density aka the cosmological constant that Einstein added to make the universe "steady state". The value of the cosmological constant is a fix to an otherwise dynamic universe that naturally wants to expand or contract, but will never be flat for more than a passing instant.