English is not my first language, so I may be wrong, but the way I'm interpreting it is that okay, yeh, one character is bearing The One Ring from the start (it doesn't state, that he begins the game as a Ring-bearer). However then (not actually "then", because it happens simultaneously, but just for the sake of an argument), he "gains" the keywords Ring-bearer and Ring-bound and "becomes" the Ring-bearer. I think "gaining the keyword Ring-bearer" and "becoming the Ring-bearer" are two phrasings of the same thing in this case. So in my head, what would happen is the following. Frodo would be able to start the game with the Ring, but would then be unable to gain the keyword Ring-bearer and would therefor not be a legal pick.