The balancer at Discount here uses the center hole. I would imagine a lug nut adapter is there for center holes that are out of round. I've just never seen one used.

 

The balancers I've checked out use a tapered cone that centers the wheel. The cone rides on the balancing taper in the center of the wheel. This taper in the wheel is not affected by an out-of-round center hole because it is a 45 degrees. The cone fits it and holds center regardless of the center hole imperfection as long as there is still some of the taper left on the wheel.

I have had two sets of H2s on this truck and a set of six on my trailer. Four sets of tires balanced and then re-balanced after they had a lot of wear. Never a balance problem related to the center hole and I cut them myself with a hole saw and a router.

The center hole is just a clearance hole for the hub to come through. It carries no weight.

 

Right on, this is the correct description. As long as you still have the chamfer then the through hole is no issue.

Tires must be the issue. I've seen one set of tires (same manufacturer, size, etc.) do all sorts of weird pulling and balance things while another set is just fine...