Cosmology Principle vs the actual center of the Universe

The first evidence I can come up with quickly is General Relativity.If Earth is the center of the Universe then everything orbits the Earth. Having something as massive as the Sun orbit this tiny speck of dust called Earth violates GR. If we consider the center, then, to be the Sun then the same relation of the 200 billion+ more massive Galaxy to the dust speck of Sun violates GR. Same then for Galaxy to Local Group, Local Group to Cluster, Cluster to Super Cluster, etc.An actual center could not be said to move. In relation to the more massive structures in the Universe we move and GR dictates this relative motion.