You forgot a little item called the supremacy clause of the Constitution which gives the Federal law jurisdiction over the states or local governments-especially on issues like civil rights. Our entire system relies on the integrity of the voting system and the principle of one man, one vote. One miscast fraudulent vote can negate the vote of another lawful citizen and deprive folks of their equality of constitutional rights. 
After all, if the Federal government could not intervene in such issues, blacks would still deprived of their voting rights in the South. 
And if the fifty states were allowed to run their own elections without any oversight by the Justice Department we would have 50 little fiefdoms running sham elections and corrupt governments. 
Hmm...sort of like Hawaii government now that I think about it.