Also, “Christ is King” is one of those things you hear so frequently they’ve lost some meaning. Fresh in its time, it was more clearly understood that if Christ is King, no one else can be.
This is why the establishment Jews that were in power at the time were killing Jesus’ contemporaries (Matthew 2:1-18) and trying to convince Rome that the Christian movement was a threat (John 19:12), and why the chant when Pilate offered Jesus’ release was “we have no king but Caesar” (John 19:15)
It’s true that the Christian movement recognized Jesus’ authority and that that is a threat to Roman authority, just not in the way it was being pitched. Usually that kind of thing goes along with a violent insurrection. But the kingdom of God was to grow in a different way, through service and love.
More on how states are a rejection of God’s authority: 1 Samuel 8