I think the order in which candidates for one office are listed is a separate issue from "down-ballot" candidates, which refers to the order in which different offices that are up for election are listed. The town park committee may be a down-ballot office, but that doesn't imply anything about the order in which those devoted citizens who want to be on that committee are listed. That said, I've heard the same thing: among the candidates for a single office, being listed near the top improves one's chances of election.I haven't heard the expression, but it's a proven fact that for each office, the candidate highest up in the list, has the best chance of being elected.
When election officials tried to be non-partisan and list candidates alphabetically, it resulted in candidates with surnames beginning with a, b, c, etc having an overwhelming advantage.
Nowadays, U.S. election ballots are printed in multiple variations with candidates sorted randomly.