"Gateway Ancestor" in this context has a specific meaning - for an American who is descended from British royalty, it means the emigrant ancestor who came from Britain to North America. Various compendia (the best-known of which is Douglas Richardson's Royal Ancestry and its various predecessors, including Plantagenet Ancestry) trace the descent from royalty to the gateway ancestor, and then leave it up to the American genealogist to establish the descent from the gateway ancestor to yourself.
In your particular case, it appears that the claimed gateway ancestor is James Elzie Moore, who was colonial governor of [South] Carolina 1700-1703 and died in Charleston 1706.
I don't see any mention on James Elzie Moore's profile (Moore-10153) of him being a Gateway Ancestor. He is also not in the list of Gateway Ancestors from Edward III in the free-space page cited by Will Palmer above, which appears to be derived from Richardson's Royal Ancestry. So I believe the task is to see whether Richardson or others would be interested in doing the work to confirm him as a Gateway Ancestor. I should note, though, that there seems to be a high probability that this will not succeed, as the Research Notes on James Elzie Moore's profile suggest that his parentage is actually unknown, with at least three different theories being proposed.