The union is simply an upper bound of the chain. What if the chain had one element, say $\{I\}$? The union of this chain is just $I$. So why is $I$ a maximal element now?
The only condition on the chain itself which ensures that its upper bound is a maximal element is to require that it is maximal. But now you're working around yourself: requiring that every partial order has a maximal chain is equivalent to Zorn's lemma.
What Zorn's lemma ensures is that by checking a local condition (i.e. that chains have upper bounds), we can derive a global property of the partial order (i.e. the existence of a maximal element).
If you want to understand better the use of Zorn's lemma, I suggest thinking about the Teichmüller–Tukey lemma instead:
We say that a family of sets $\scr F$ has finite character if $A\in\scr F$ if and only if every finite $A_0\subseteq A$ is also in $\scr F$.
(Teichmüller–Tukey) Let $\scr F$ be a family of sets with finite character. Then $\scr F$ has a maximal element under $\subseteq$.
To see why this is "the usual use of Zorn's lemma", note that the standard appeal to Zorn's lemma is in the case where a certain property is of finite character. For example, in the case vector space bases, being linearly independent is a property that has finite character: if a set is not linearly independent, there is a finite subset witnessing that.
The only reason to use Zorn's lemma is that sometimes it's slightly simpler. Like in the case that you want to extend an ideal to a maximal ideal, or even extend a given linearly independent set to a basis. But the idea stays the same, we use the finite character of our property.
Best Answer
"Every unital ring (other than the trivial ring) has maximal ideals" is equivalent to the Axiom of Choice (and hence to Zorn's Lemma) in ZF. So one cannot prove it without Zorn's Lemma or some equivalent statement.
"In a unital ring $R$, every proper ideal is contained in a maximal ideal" follows from the first proposition by taking the quotient $R/I$ and lifting a maximal ideal using the lattice isomorphism theorem. Conversely, if in a unital ring every proper ideal is contained in a maximal ideal, then every unital ring has maximal ideals: just find a maximal ideal that contains the zero ideal. So this proposition is equivalent to the first, and hence cannot be proven without Zorn's Lemma or some equivalent statement.