As you point out, it is a mistake to confuse the state with the church. Politics is a distraction and an idol. I take "give unto Ceasar what is Ceasar's," to mean we give a moderate amount of attention and emotional energy to poliitics--just enough to participate in democracy. But we place our hope and trust in God.
"To turn your back on politics is to turn your back on your neighbor, a failure of love."
I'm not seeing how that follows from what you've written.
As you point out, it is a mistake to confuse the state with the church. Politics is a distraction and an idol. I take "give unto Ceasar what is Ceasar's," to mean we give a moderate amount of attention and emotional energy to poliitics--just enough to participate in democracy. But we place our hope and trust in God.
Those who live by politics will die by politics.