Retraction of a Riemannian manifold with boundary to its cut locus
Let me continue the comments above here so I can include a figure. Here are examples of the medial axis of two different convex polygons (from my own work):
The term medial axis is used in computer science to denote the same concept as the cut locus.
Franz-Erich Wolter wrote his Ph.D. dissertation on "Cut loci in bordered and unbordered Riemannian manifolds" (Technische Universität Berlin, 1985). That might contain some useful information.
The 'simple lemma' certainly isn't always true. Let's consider just open disks in the Euclidean plane with a flat metric. Fremlin (Proc. LMS, 1997) calls the set of points without a unique nearest boundary point the "skeleton". He gives an example where the skeleton is somewhere dense (that is, its closure, which you call the "cut locus", has nonempty interior). The skeleton will always be an R-tree.