Do not suppose that I came to cast (βαλεῖν) peace over the earth. I did not come to cast (βαλεῖν) peace, but a sword. (Matthew 10:34 DLNT)
The Disciples Literal New Testament accurately describes how Jesus was not going to bring peace. A meaning of the word βαλεῖν means to cast or throw without caring where it falls. [906-ballo]
Jesus did not come to throw out peace without caring where it falls. Rather He came specifically that those who believe in His Name would have the peace He brought to the world:
“I leave you peace. I give you My peace. I do not give to you as the world gives. Do not let your heart be troubled, nor let it be afraid. (John 14:27 DLNT)
“I have spoken these things to you in order that you may have peace in Me. You have affliction in the world, but take-courage— I have overcome the world”. (John 16:33 DLNT)
In other words, His peace would be implanted one person at a time. The internal logic of Matthew 10:34 is that Jesus did not come to cast His peace without caring where it fell, rather He came to cast His sword (the Word) without caring where it fell.