Re: "why have a system that allows voting in more than one place in the first place?"
In which case the post-election checking that such names have only been ticked off once will show it up
The number of accusations of such activity are usually 10-20 times higher than it actually happening, but every accusation is investigated no matter how many times the accuser has already been shown to be a foaming looney
When it does happen, it's almost invariably multiple voting for the conservative/rightwing candidate - but more to the point the number of occurences is so low that it makes no difference to the result. Additionally: It's almost always detected by the system working as designed rather than any 3rd party accusations
As I pointed out elsewhere, postal voting fraud is an entirely different kettle of fish, but it's fairly clear that the loudest accusers of such behaviour are trying to distract from postal voting issue because they're projecting their own behaviour onto other groups as a deflecting tactic
In both New Zealand and Australia, one can vote in ANY polling station in any part of the country and it's been this way since before I was born. There's no such thing as an "assigned polling station" for a voter and the expectation is that you can rock up to any station in your district (and usually the neighbouring ones) without having to cast a "special vote" (which is cast as normal but put in a sealed envelope and sent to the returning officer for the district in question after voiting closes. If there is a question about voting eligibility(*) the same procedure is undertaken - the vote in question is placed in a sealed envelope and the question of eligibility sorted out AFTERWARDS (if eligible, the envelope is opened and counted, If not, it's destroyed unopened)
(*) Such questions usually revolve around correct entries on the roll. It's mandatory to register in most countries and not being on the roll on the day doesn't preclude being allowed to vote if it's determined that someone should have been on the roll (ie: was eligible)
Americans are amongst the worst for active voter suppression activities - something that's usually only seen in tinput dictatorships elsewhere in the world and mostly advocated by fascist wannabes who are afraid they might be kicked into the weeds if everyone who should be voting CAN vote
Don't forget: Hitler and co never managed to exceed 30% of the votes cast even when they were the only party on the ballot
For your amusement: One university I attended used to have "No confidence in any candidate" as an option in student body elections. That was removed the year after "No confidence" managed to get twice as many votes as every other candidate combined. It's made me muse about setting up a political party with that name and see what happens