I am looking for examples of results which may be proven without resorting to the axiom of choice/Zorn lemma/transfinite induction but whose proof is quite simplified by the use of the axiom.

For example, in the Poincare-Bendixson theory, it is shown that any closed orbit of a planar vector field contains a fixed point. This is done first by showing that any closed orbit contains a closed orbit. Then it is straightforward to conclude using Zorn's lemma. The conclusion can be reached without resorting to Zorn's lemma, for example by looking at the area bound by a closed orbit. But this is not so straightforward. One has to give a meaning to such area and then show amongst other things that this area depends continuously on the orbit.