The last remark first:

This gives me the feeling that Monte Carlo is fairly valid. (How's
that for rigor?)

Not very ;-).

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 still can create a pointy tail; just as pointy as the front. My
experience is that the tail is much more blunt than the front. That there
are already more than a single configuration at maximum distance makes that
reasonable.