[It is the U.S. Constitution and Bill of Rights that makes election the business of the federal government. By giving all citizens the right to vote.]
Not entirely true.  The Constitution is oddly silent on exactly _who_ gets to vote (except for defunct 3/5ths vote for slaves) or how it will be regulated.  That is left to legislation and originally it was left to states to regulate.  Every law (not Constitutional amendment) that provides for voting (Jackson and universal male sufferage, Afr.Amer. male voting from Civil War era, Women's vote early 20th century, etc.) were a result of when states demonstrated a particular lack in regulating voting appropriately.  In other words, it isn't any Constitution provision or amendment that provides for federal regulation of voting but an history of poor judgment by the states in regulation of voting.