"People use armed force all the time with their neighbors."

This was not phrased correctly. It should have said, "used," and it should have included the threat. Back when governments were weaker and there was more crime and fewer educated people, courts weren't great at punishing violence, so violence and the threat thereof were widespread - look at the historic power of organized crime. This remains true today, but it's less true because our ability to enforce the law has improved.

People used violence when it was effective and they wouldn't get in trouble with other, more powerful people. Nations used violence when it was effective and they wouldn't get in trouble with other, more powerful nations. Nuclear weapons simply changed how much trouble they'd get into, just as better law enforcement monitoring and a lower general crime rate, and falling cultural acceptance have changed how likely people are to get into trouble when they use force. I really don't think the two examples are as different as you claim they are.