If there really was a force similar to neutron binding energy but leading to a more dense state, can you explain why the density of large black holes gets less and less?

Come on .... really ... you are a mathematician.

The density goes down if you consider it from the event horizon which is what you are doing because that is special to you.

So your excercise put a dot on a page and write a solar black hole mass. Now calculate the radius of the event horizon.

Now double the mass you wrote for the dot and calculate the new event horizon radius.

When you have done that now write the spherical volume for each sphere at the event horizon.

Surely you see what is happening as a mathematician the event horizon is going out as the square or the radius but the volume is going as the cube of the radius.

The reality is all the mass is in a single point but viewed from the event horizon the density is going down as the volume is a cubic .... good old 4/3 * PI * R^3

How hard is that to understand for a mathematician, I would have thought you could work that out yourself.

Density is an AVERAGE that is the problem and you have nothing like a uniform spacing. That actually directly shows you insignificant the Event horizon truely is in that it gives you that stupidity.