I think Myers only considered analytic metrics, see his papers "Connections between differential geometry and topology I and II", Duke Math. J. 1 (1935), 376-391, and 2 (1936), 95-102.

For arbitrary metrics on $S^2$ the cut locus is indeed a tree. This can be deduced from e.g. in [Shiohama and Tanaka, Cut loci and distance spheres on Alexandrov surfaces] who work in the (more general) settings of Alexandrov spaces homeomorphic to surfaces and prove that the cut locus is a local tree.
(This paper is available online, search by title). Specializing to the case when the surface is a Riemannian sphere we observe:

The complement to the cut locus is a 2-disk.

If the cut locus contains an embedded circle, then by Jordan curve theorem the circle separates $S^2$ into two disks, and by part 1 one of the disks must lie in the cut locus, so the cut locus cannot be local tree.

One should be careful with what is meant by a local tree. By a result of Gluck and Singer [Scattering of Geodesic Fields II, Annals of Mathematics, Second Series, Vol. 110, No. 2 (Sep., 1979), pp. 205-225] there is a positively curved Riemannian metric on $S^2$, in fact a convex surface of revolution, for which the cut locus cannot be triangulated, so it is definitely not a finite tree.

For recent works in this area see papers of Itoh, e.g. http://arxiv.org/pdf/1103.1758 and references therein.