Re: in the UK at least
Voting in the UK is simple because we only vote on one thing at a time. Each ballot gets its own piece of paper, and they are counted in two steps: first sort the paper into piles according to who they vote for, then count the number of pieces in each pile. Both steps are easy to do in parallel, so if you use counting staff proportional to the voting population it happens in constant time. It's also relatively easy to spot-check that no votes got into the wrong pile, or that each pile has the reported count. Basically, it scales well.
In America they vote on vastly more things. Not just president, but elected officials at various levels. Instead of each vacancy getting its own piece of paper, they combine them all onto one sheet. This makes it impossible to count manually using the UK method. That's why they use automated systems of various designs. (Not saying their system couldn't be adapted, eg by using perforated paper and splitting up the sheet into one strip per vote. Historically they've not done that.)
I've no idea whether the Swiss are like the UK or like the US.
Biting the hand that feeds IT
