The Golden Rule is about meeting people where they are. That means using language they understand, situations they are prone to experience, everyday situations, etc. Loving your neighbor as yourself means learning about your neighbor, then lovingly tailoring your interactions with your neighbor in ways based on your accumulated knowledge of them.
When someone really pays attention to you and shows you how well they see who you are as a person, that is a display of love. You feel loved when that happens. Everyone does when they feel truly seen and accommodated.
That said, Jesus's use of those terms like peace and sword in Matthew 10:34 is largely symbolic, although the use of peace is fairly applicable.
The sword Jesus brought was the sword of truth. St. Michael knows a thing or two about those from what I hear. If a person, say...a Pharisee at the time of Christ...was a total egotist who thought his mommy and his daddy were the super bestest human beings in the history of ever and that he was the pinnacle of human evolution...studies the Torah super hard...Karens everyone about pedantic law violations...boisterously offers sacrifices at the temple so everyone notices...all that "look at how pious I am" stuff.
People like that are not going to respond well when they are told they are preaching the wrong stuff. They're going to feel drawn down upon. They're going to feel like you're pulling your sword when you correct their teachings. They're going to feel like you've violated their peace of mind. They're going to feel like you're subverting their status in society and that threatens them at a deeply psychological level.
Pretty much it's only sigma personalities who are both competent enough and apathetic enough to roll solo. The apathy is directed towards the pack's (society's) bullshit hierarchy games. Sigmas don't care to play that. Everyone else plays for their own reasons. Most of them play because they're too incompetent to roll solo so the prospect of being booted from the pack constitutes a serious degradation of their prospects for survival, succession, and general flourishing.
So, psychologically, a lot of these people literally felt fight or flight responses when confronted by Jesus regarding their teachings. Their perspective that Jesus was "drawing down upon them" with a sword was essentially in the same category, neurologically speaking. The same neurochemicals are pumping in both cases.
Swords of truth have a tendency to garner reactions like that. Mark Twain understood that when he said "It's easier to fool people than to convince them that they have been fooled.” Likewise, it's easier to get people to believe false things than it is to convince them that they already believe false things. The bread and butter of the ego is the sunken cost fallacy.