I do not know if this is an example, but it seems related:

Michael Roddy (and others), if I remember correctly, have proven results using the fixed point property (FPP) for posets that seem to enable them to deduce the existence of maximal elements. It may be, though, that those results are themselves based on results that use Zorn's Lemma (namely, Anne Davis's result that a lattice has FPP only if it is complete, a converse to Tarski's Fixpoint Lemma).

See, for example, Corollary 2.4 of Michael S. Roddy, "Fixed Points and Products: Width 3," Order 19 (2002), 319-326.

I do not know if this is an example, but it seems related:

Michael Roddy (and others), if I remember correctly, have proven results using the fixed point property (FPP) for posets that seem to enable them to deduce the existence of maximal elements. It may be, though, that those results are themselves based on results that use Zorn's Lemma (namely, Anne Davis's result that a lattice has FPP only if it is complete, a converse to Tarski's Fixpoint Lemma).

See, for example, Corollary 2.4 of Michael S. Roddy, "Fixed Points and Products: Width 3," Order 19 (2002), 319-326.