I think it's the "scale" of the aspirin. We used the same sort of method occasionally when I was a kid and raised pigeons. Every once in a while, we'd have a pigeon that would need to be "put down" (injured too badly to survive, serious illness that could infect others, etc.). The easiest way we found was to simply push an aspirin down the throat of the pigeon. Like Michael said so well, they "sleep" forever.
Really, it's an overdose. Three aspirin might not hurt a large dog, but they might kill a lap dog. People can overdose on aspirin and die, too; it just takes a lot more than three at a time for it to be lethal.