Sort of similar to R. Moshe's response, I heard from a noted Rosh Yeshiva that the optimal (and indeed traditional) authority structure, was a pyramid. On the bottom was each person's personal rabbi; usually the rabbi of the town. When those rabbis had questions they couldn't answer they forwarded them to someone they felt was better qualified. This process continued until the questions reached the rabbinic elite at the top of the pyramid. Nobody appointed them, rather rabbis seeking better qualified rabbis inevitably formed such a pyramid where questions reached those qualified to deal with them.
In such a system (he stressed) the individuals who first ask the questions have no need to investigate who lies at the top; each level interacts with the level above it.
Accordingly we really don't need to worry about who the gedolim are. We just need a qualified rabbi who knows us, who we trust to forward difficulties to someone better qualified if need be.