A ground connection shorted to ground won't do anything bad. But if you're getting ANY resistance reading between the lug on the starter switch and the body of the switch, that's a sign that something inside the switch is partly shorted to ground. Your multimeter doesn't have the capability to push voltage thru the connection that the battery does, which would explain your small amp draw. So it sounds like the starter switch is the problem. Have you removed the switch and inspected the workings of it. It could be rust between the body and lug, either inside or outside. Or you could just get a replacement switch and see what that does.
