Do not think I have come to bring peace to the world. I have not come to bring peace but a sword.

Jesus goes on to talk about families being turned against one another. He admonishes us to follow Him no matter what, no matter who it might offend.

The way I understand this is His followers would upset family members and friends because their beliefs would be questioned and challenged by those of the Kingdom of God. So families and friend groups etc would be turned against one another.
Do we not also see this today? The new rise of "spirituality" (paganism) is clashing again with Christian thoughts and practices. We are seeing families turned against one another. So what are we called to do? Follow Jesus, do the work of the Kingdom even in the face of temporal strife.

The happenings of the last week have stirred this in me. As we see the demonic play out in the celebration of murder we should not be surprised when we see it in our own families. Of course pray for them. Lovingly direct them to the One True God and the Truth. But keep following and leaning into Jesus and let the Spirit do His work.

I'd caution trying to directly apply Bible verses like these to our times. Yes, everything said by Jesus has a timeless quality, but I wouldn't treat it as some sort of prophecy that only applies to our current society. We're on the verge of a major social upheaval, and everyone fighting everyone is typical of these times throughout history.

Related musings: In Biblical times, and in much of the world today, people are tribal. Rights, responsibilities, benefits, and punishements all flow from this. The old saying, "Me against my brothers; me and my brothers against my cousins; me, my family against my trible; my against my confederation; my confederation against he world" applies. Prior to the modern age, law was a much more abstract concept. If you were an orphan in medieval Europe you often had no rights at all. Anyone could abuse you in any way, because you didn't have a family to stick up for you. This is completely different in our society. You can't beat up a homeless person on the street, because the government defends everyone to some extent. This all started with the early law codes, but they didn't quite work the same way.

Imagine a beggar starts grabbing fruit off the ground of your orchard (this is a legal, Biblical right for them to do). Now your uncle shows up, gets mad, and starts beating up the beggar. You and your brother are the only witnesses. Under Biblical law, at least two witnesses are required to convict your uncle of the crime he clearly committed. So what is the right thing to do? Do you support the beggar when he goes to the judge with his complaint? Or do you protect your uncle? For non-JudeoChristians this is easy. You keep your mouth shut and protect your family. For us, we are commanded not to bear false witness and to give fair justice to the meek and lowly. So a follower of Jesus should turn against his family in the interest of higher, more Godly principles.

In Jesus' teachings doing the right thing is treating everyone like you would treat your brother. That means standing with them against your cousins, your family, your tribe, and your confederation. That is guaranteed to cause conflict with people that don't follow the same teachings

Yeah, I agree, and you're not misapplying anything. Nowadays I'm just hypersensitive to people saying the Bible was written specifically for our current situation, and you're not doing that.