Why do group II elements conduct?

Even when an isolated atom has a filled shell, the electron bands in a solid crystal may be partially filled. The reason is that bands that originate from different atomic orbitals may actually overlap in energy. This is shown on the left in the figure below.

When the bands overlap, the lowest-energy state has some electrons displaced from the top of the lower band to the bottom of the upper band, so that the Fermi energy in both bands is the same. This leaves partially filled bands, enabling conduction to occur.

Your intuition is probably biased by one-dimensional periodic potential examples of band structures, where it is impossible to have a band level crossing (it would imply the existence of more than 2 independent solutions of a second ordere ordinary differential equation).

In more than 1D things are different and energy levels corresponding to more than two k-points are possible. This is the case of second column elements and also explains the possibility of a range of behaviors for the elements of group IV (Carbon is an insulator in the normal pressure diamond lattice, while Lead is a metal).