Suppose a aboriginal guy from the Amazon wakes up naked on a snowy Manhattan street... He could be the best Tapir hunter in the five valleys, but that doesn't count for much if he inevitably either freezes or gets locked in a cell for indecent exposure.
But that aside, I think Arnold was saying something misleading: When people call him a self-made man, they're not saying he's not the benefactor of the efforts of the people who came before him. Even the aborigine is on the receiving end of 15,000 years of tribal knowledge about how to survive in the American wilderness. What they're saying is that he's not one of the people who was handed success by default, and it's fine to say "thanks" to the people who helped you in your efforts to better yourself, but there's still an important distinction there.