Isn't proving $A$ iff $B$ iff $C$ by showing $A\to B$ and $B\to C$ and $C\to A$ circular?

Yes, you are absolutely right the proof $A \implies B \implies C \implies A$ only shows that $A,B,C$ are equivalent. So if one of them is true, all the others are true and if one is false all the others are false. However, the proof you linked doesn't try to show that the principle of mathematical/complete induction is true or the principle of well ordering is true, it only shows that they are equivalent. In fact, in the usual framework of mathematics these are taken as axioms. The proof shows that you only need to assume one of them as an axiom and you get the other for free.