McKay's secret probably would have cost him the election
Near the beginning of the movie, Marvin Lucas and Bill McKay are discussing McKay's candidate possibility at his southern California home. McKay's wife, who is also present, mentions that he will have to register to vote first. Lucas responds that this should be kept secret.
Most likely, the public would inquire about McKay's past voting history, and his failure to vote in previous elections would be considered scandalous. I believe the press would hound McKay relentlessly for information regarding his party affiliations, and exploit the lack thereof. Nevertheless, nothing is mentioned further about this issue in the movie.