This question quotes and exercise from a book.

It says that it is an exercise whose purpose is to enlighten the reader as to "what transfinite induction is and why it is always replaced by Zorn's Lemma".

This seems incorrect to me.

I don't think that it would be better to ignore transfinite induction and use ZL instead of it.

I agree that in some cases it might be straightforward to transform proof by transfinite induction to a proof using Zorn's lemma and vice-versa.

But I don't think that this is always the case.

Moreover, ZL requires Axiom of Choice, transfinite induction does not. (Of course, in the proof by transfinite induction AC might occur, but the theorems about transfinite induction/recursion are basically theorems saying that induction works on well-ordered sets and they do not depend on AC.)