Cross-entropy is not the KL divergence. There is an additional term in cross-entropy which is the entropy of the data distribution (i.e., independent of the model). So, you're right in that minimizing one is equivalent to minimizing the other.
Yes, you are totally correct, but I believe this term is omitted from the cross-entropy loss function that is used in machine learning? Because it is a constant which does not contribute to the optimization.
Please correct me if I'm wrong.