I think of Zorn's Lemma as a tool for efficiently presenting certain proofs whose intuitive content consists of transfinite induction. (I think people like Zorn and Kuratowski would have agreed with this.)
In more detail: The natural (for me) proofs of all six of the theorems you quoted are proofs by transfinite induction, and I believe these are the proofs people would have given in the early days of set theory. It turned out that many such proofs (like the first three of your six) used transfinite induction in the same way, and a very simple way. Rather than repeating that simple argument over and over, people abstracted it as Zorn's Lemma, a lemma that can simply be quoted every time that particular argument is needed.
A side benefit was that, for these simple arguments, people no longer needed to learn about well-orderings and transfinite induction.
But, as you observed, Zorn's Lemma abstracts only some, not all uses of transfinite induction. In principle, any application of the axiom of choice could be based on Zorn's Lemma, since Zorn's Lemma implies the axiom of choice. But that's "in principle"; it would just make the natural proof longer by prepending a proof of AC from Zorn. It's entirely contrary to the purpose of Zorn's Lemma, which is to simplify certain proofs, not to make proofs longer and harder.
In your last three examples, the usual proof uses transfinite induction in a way that's different from (and a bit more complicated than) the way that Zorn's Lemma is designed to abstract. So those proofs should be left in the form of transfinite inductions.
Even when a proof uses Zorn's Lemma, I tend to visualize it as a transfinite induction that just happened to be of the sort that Zorn's Lemma can efficiently abbreviate.
By the way, there is a version of Zorn's Lemma, which I've seen attributed to Kneser, that might be able to abbreviate more transfinite induction arguments. Instead of requiring upper bounds for all chains, Kneser's version requires them only for well-ordered chains. (If you're thinking, "Wait, that makes no difference, because every chain has a cofinal well-ordered subset," you're right about the existence of cofinal well-ordered subsets, but that fact depends on AC.)
Finally, I should point out, in connection with the last of your six examples, that, although the natural construction of an undetermined set of reals uses transfinite induction, there's an alternative proof that works nicely with just Zorn's Lemma: If $U$ is any nonprincipal ultrafilter on $\mathbb N$, then the following game is undetermined. The two players alternately "take" finite subsets of $\mathbb N$, subject to the constraint that any set they take must be disjoint from all the sets previously taken (by either player). The first player wins iff the unino of the sets he takes is in $U$.