Classical reason for good heat conductance in diamond?
I think that there is no really good classical reason.  Heat conductivity in a solid depends on bonding in that solid and bonding is inherently a quantum mechanical phenomenon.

For instance, recall that graphite is a good conductor of heat (and electricity) in two dimensions and a fairly good insulator (heatwise and electrically) in the third.  This is due to the layered structure in graphite and that is due to the bonding in graphite.