Does $\mathsf{DC}$ imply $\mathsf{AC}_ω$, uniformly?

The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice $\mathsf{AC}_\omega$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while $\mathsf{AC}_\omega (X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\mathsf{DC} \Rightarrow \mathsf{AC}_\omega$. We study for which sets and under which hypotheses $\mathsf{DC}(X) \Rightarrow \mathsf{AC}_\omega (X)$, and then we show it is consistent with $\mathsf{ZF}$ that there is a set $A \subseteq \mathbb{R}$ for which $\mathsf{DC} (A)$ holds, but $\mathsf{AC}_\omega (A)$ fails.