Why doesn't Zorn's lemma apply to $[0,1)$?

As a partial order $[0,1)$ has no upper bound. Sure $1$ is an upper bound of $[0,1)$ in $[0,1]$ or in $\Bbb R$. But that is not the same partial order. You are not allowed to go to larger partial orders when you apply Zorn's lemma.

So $[0,1)$ has many chains without upper bounds. E.g. $[0,1)$ itself.

You are actually not using Zorn's Lemma which states that if each chain in $[0,1)$ has an upper bound, then $[0,1)$ has a maximal element. However, the chain $\{1- \frac 1 n \}_{n \in \mathbb N}$ has no upper bound in $[0,1)$.