We might as well use the axiom that “you can make arbitrary choices at each stage of a transfinite induction”. It’s mostly pedagogical tradition to make students translate that into one of the classical forms of Choice in their proofs.

Does transfinite induction without the ability to make arbitrary choices make sense? Or is the axiom just that you can do transfinite induction?

Zorn's lemma is specifically that there is a maximal element of an ordered set that is structured in a specific way. That seems a lot weaker to me than just being able to make a completely arbitrary choice. Am I wrong about that?

You are correct that it SEEMS that way. They are, in fact, equivalent under traditional mathematical hypotheses about how sets work.