There's some inherent flaw here - as 'pegging' the size of unit cube to its 1-dimensional parameter, the length, does not make the volume to vanish.
Unit cubes are constrained such that all of their measurements are unit length---something which ensures that their volume is constantly 1. They do this by getting spiky, as xyzzyx mentioned. The distance to cross them shrinks as the vertex distance grows.
So they start to look totally different in high dimensions, evidenced by the ball's vanishing measure.
> And so pegging the size of our unit ball to its 1-dimensional parameter, the radius, causes its volume to vanish.
Funny. I was just going to enter a comment how that was the most useful intuition in the whole thread.