Let $X$ be a compact metrizable space and let $\mathcal{K}_{ne}(X)$ be the collection of non-empty closed subsets of $X$ with the Vietoris topology (i.e. the topology induced by the Hausdorff metric for any compatible metric on $X$).

Question:When does there exist a continuous function $f: \mathcal{K}_{ne}(X) \rightarrow X$ such that for every $G \in \mathcal{K}_{ne}(X)$, $f(G) \in G$?

This feels like it should have been studied before, but I am unable to find a reference.

Some easy observations:

If $X$ has a continuous choice function for non-empty closed sets and $Y$ is a closed subspace of $X$, then $Y$ has a continuous choice function for non-empty closed sets.

$\inf : \mathcal{K}_{ne}([0,1])\rightarrow [0,1]$ is a continuous choice function for non-empty closed subsets of $[0,1]$. So we also have this for any closed subspace of $[0,1]$, such as Cantor space and any countable compact metrizable space.

The circle and the tripod (three copies of $[0,1]$ glued together at $0$) both do not have continuous choice functions for non-empty closed sets (in both spaces given a set with two points there is a continuous path that makes the points switch places while keeping them separate). So no spaces in which these embed do either.

Any finite disjoint union of spaces with continuous choice functions for non-empty closed sets also has a continuous choice function for non-empty closed sets (having elements in a clopen subset is a clopen condition in $\mathcal{K}_{ne}(X)$, so we can patch together the choice functions by cases).

A reasonable conjecture is that any such space embeds into $[0,1]$, but I could also see something tricky like the pseudo-arc having a continuous choice function for non-empty closed sets.