Romans 13:5 is a near echo of a saying of Jesus: “Therefore, it is necessary to submit [live within the order] to the authorities … because of conscience.” Here’s what Jesus said in Matthew 17:26-27: “Then the sons are exempt [from Temple tax] … but so that we may not offend them, go to the lake… give it to them for my tax and yours.”
If Paul brings in conscience after his argument from pragmatics — they’ve got the sword (Rom 13:1-4), does that mean conscience is secondary?

That is, the ultimate reason Christians submit to the authorities is because they live before God. Christians know God is Sovereign; they know God is ordering the world; and they want to live within God’s order. It is the same argument we find in 1 Peter 2:11–3:12, esp enunciated in 2:13-17. Paul (and Peter) urge the Christians to be good citizens, not rebellious and seditious — why? Because government is part of God’s good orderliness in this world.
Always? Obviously not. In general? Obviously.