No, this principle implies $\mathrm{DC}(\mathbb{R}).$ Suppose $T$ is a tree on $\mathbb{R}^{<\omega}$ with no leaves or branches. Let $\mathcal{A}$ consist of all $A \subset \omega \times \omega$ such that for some $\langle r_i: i<n \rangle \in T,$ $$A=\{(2i, j): i < n, j \in r_i\} \cup \{(2i+1, k): i<n, k \not \in r_i\}.$$

Then $\mathcal{A}$ has no infinite chains or maximal elements.

This generalizes to proving $\mathrm{DC}_{\kappa}(\mathcal{P}(\kappa))$ for all $\kappa,$ so, over a base theory of ZF - Foundation, this proves the pure axiom of choice, i.e. that the well-founded universe satisfies AC. In fact, the principle is equivalent to pure choice since it lives inside the well-founded universe.