Longitude on the sphere is uniformly distributed. Latitude isn't. If you chose the angle in [0, pi) (the "colatitude" - https://en.wikipedia.org/wiki/Colatitu

Longitude on the sphere is uniformly distributed. Latitude isn't. If you chose the angle in [0, pi) (the "colatitude" - https://en.wikipedia.org/wiki/Colatitude https://en.wikipedia.org/wiki/Colatitude) from the right distribution this would work. See for example https://en.wikibooks.org/wiki/Mathematica/Uniform_Spherical_Distribution https://en.wikibooks.org/wiki/Mathematica/Uniform_Spherical_... - it looks like selecting 2 arcsin sqrt(X), where X is uniform on [0, 1], does the trick.