“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.
Distribution is what we are talking about.
You sound like someone who is familiar with statistical analysis, so I urge you to start looking at the raw data published by the various election offices.
At the precinct level the actual count for any candidate has a random component, since the number of eligible voters in any precinct has a random component, and the number who turn out to vote does to.
As one example, I would expect that the turnout percentage by precinct would be a distribution around a mean, and certainly not a fixed, constant value.
The percentage of voters who choose a particular candidate in any precinct should also have some variability.