Which compact metrizable spaces have continuous choice functions for non-empty closed sets?

It's an old (1981) theorem by Jan van Mill and Evert Wattel (see this paper) that a compact space has a continuous selection iff it is orderable. (So has a linear order whose order topology is the topology on $X$). $F \to \min(F)$ and $F \to \max(F)$ are then the two only continuous selection functions IIRC. Even a continuous selecting function for $[X]^2$, the subspace of doubletons, is enough to get orderabilility.

If a space $X$ admits three distinct points $x_1,x_2,x_3\in X$ such that $X\setminus\{x_i\}$ is connected for every $i=1,2,3$, then $X$ does not admit a continuous choice function from the set of two-point subsets of $X$ back to $X$.

Indeed, such a choice function could be used to define three continuous maps $f_i:X\setminus\{x_i\}\to \{0,1\}$, and at least one of $f_1,f_2,f_3$ would have to be non-constant. Contradiction.

This argument excludes the pseudo-arc.