I agree: the proof suggests that the integers are essentially different because their order is not dense; the rationals are essentially different because Alice's sequence might not converge; the computable reals are different because Bob might not be able to compute membership of S. These aren't set-theoretic properties, I suppose, but they're certainly interesting. Of course, the mere failure of an argument is not proof of the negation, but I'd say the proof is illuminating. By contrast, Cantor's diagonal argument is laser-focused on proving this one fact (and it does so well!): it's a single technique, pared to the bone, telling you exactly what you wanted to know and no more.
Yes. Though it's also interesting to see how Cantor's proof can fail.
Eg you can try to apply Cantor's proof on the list of all integers (written in decimal form) to attempt to prove that the integers aren't countable. Or on a list of all rationals.
