I wrote a simple proof and I want to verify it.
I simply argued that the set of all totally unordered subsets forms a partially ordered set, ordered by inclusion. Every chain in this partially ordered set has an upper bound (the entire poset) so we can apply Zorn's lemma to show that our set of totally unordered subsets has a maximum.
Another word for totally unordered subset is "antichain".
Your argument is flawed, since the entire poset need not be an antichain.
Instead you need to show that if $\mathcal C$ is a chain of antichains, then $\bigcup\cal C$ is an antichain as well.