Kurepa trees, continuous images, and perfect set properties

Building upon work of L\"{u}cke and Schlicht, we study (higher) Kurepa trees through the lens of higher descriptive set theory, focusing in particular on various perfect set properties and representations of sets of branches through trees as continuous images of function spaces. Answering a question of L\"{u}cke and Schlicht, we prove that it is consistent with $\mathsf{CH}$ that there exist $\omega_2$-Kurepa trees and yet, for every $\omega_2$-Kurepa tree $T \subseteq {^{<\omega_2}}\omega_2$, the set $[T] \subseteq {^{\omega_2}}\omega_2$ of cofinal branches through $T$ is not a continuous image of ${^{\omega_2}}\omega_2$. We also produce models indicating that the existence of Kurepa trees is not necessary to produce closed subsets of ${^{\omega_1}}\omega_1$ failing to satisfy strong perfect set properties, and prove a number of consistency results regarding \emph{full} and \emph{superthin} trees.