The sums of the votes are random (in the sense of these kinds of probabilities) - unless the voters in a precinct get together and agree that “176 of us will vote blue and 123 of us will vote red”.
“The sums of the votes are random (in the sense of these kinds of probabilities)”
Well, that’s the thing, you have to qualify the statement. Because they are actually definitely not random at all. You can say they approximate a random distribution, but they are not random.