I realise that the colonies decided to become independent, but they might consider retaining some of our voting rules:
"could take their absentee ballot to Boss Tweed, mark it in his presence and seal the envelope in front of him, then he could give the payoff and drop it in the mail for the voter"
In the UK, absentee ballots are cast in advance of the election. Having given your ballot you are still permitted to vote at a polling station, invalidating the absentee vote. I realise that is an imperfect safeguard.
Another safeguard is perhaps of more relevance in areas where people try to frustrate voter registration. UK voters are required by law to register, although they are not compelled to vote. While we may take that rule for granted, it would be a valuable safeguard in certain parts of the US, where people try to disenfranchise certain voters.