My 2c worth as an outsider (ie not voting, not US citizen, not even in the country!):
The whole show should be organised, run, checked & paid for by an independent
federal body (ie a US Federal Election Authority). All candidates & parties pay their own campaigning & advertising costs, of course, but give a separate body the funds to actually run the enrollment, voting & counting
process. Get all the partisan groups out of it (on both sides of the political fence).
That's how it's done here in Oz - no-one EVER complains of irregularities in the process, as an independent body looks after it all, NOT the Government of the country (or state, etc).
As things are in the US, to us on the outside it often looks like you need to have the UN in there to monitor how each election is run. Very "Third World" guys!!