SOLVED: How to make the Einstein Ricci tensor coordinate independent - we do not need to

We had a confusion about tensors. In the Einstein field equations, each side is a tensor. To change to a new coordinate system we have to transform both sides. Then a solution of the field equations is a solution also after the coordinate transformation.

Our confusion was due to the fact that we believed that the stress-energy tensor is calculated using the proper coordinates of the observers inside the manifold. Then the stress-energy tensor would be coordinate independent, and a coordinate transformation would break the Einstein field equations. The stress-energy tensor has to be calculated from the coordinates, not from proper time or proper distances.

Question. There is no need to make the Einstein equations coordinate independent. But is it possible to make them coordinate independent?