The argument from Thrasymachus and middle Foot is that injustice is not always bad.
Regarding your own argument, the conclusion looks to be underdetermined. We are neither practically bound nor morally obligated to carry out every act which promotes justice, and yet this is what your argument entails. (See also my post to you here.)
For the austere Kantian it is respect for the moral law which leads one to abide by it, not choice-worthiness or goodness. Similarly, Kant would deem the person confused who praises him for doing what is obligatory. Of course, as you’ve pointed out, @Jay is very far from an austere Kantian, so there’s that.
In order to claim that it is a tautology you would have to show that “justice” and “moral good” mean the same thing. I don’t think they do, and justice being a term of moral approbation doesn’t show that it is a tautology. In fact lots of contemporary thinkers literally think that classical justice is a moral evil.