I wonder about the validity of your Monte Carlo analysis. It seems
to be based on an intuition about how fast the number of configurations
falls off with the distance from SOLVED. I share the intuition, but
I'm not sure I can rigorize it, and that makes me cautious.

What prevents a group from having a "pointy tail", that is, a "corridor"
of elements at increasing distances from the identity? In fact, does
the number of elements as a function of distance have to be unimodal?
Could this function have a "waist"? Intuitively, this sounds
impossible, but I am wondering what constraints on such functions are known.