Disclaimer: this sounds more authoritative than is intended--I really
don't know what I'm talking about.
It couldn't be very pointy. From the most distant configuration,
there are 6 positions immediately before it. There are 6^2 two steps
away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually.
This is necessarily so, as if any of the configurations reachable with
two twists (for example) are closer in than (max - 2) steps then you
could reach the maximum configuration by getting there and then doing
the two steps.
This gives me the feeling that Monte Carlo is fairly valid. (How's
that for rigor?)
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.