Charles Pate's will is what's disputing it. He has three sons in the census records. He names three sons in his will. The three sons are John, Charles, and Zachery. That is a documented fact that can't be disputed. Charles Pate of Robeson County, NC did not have a son named Abimelec.
I haven't seen any evidence for his place of birth but I have seen plenty of evidence that he isn't the son of Charles Pate of Robeson County no matter how much you may want it to be so.