The cartesian field {x, y, z} is not general, it is radial. No wonder that in spherical coordinates it has zero angular components. The documentation of TransformedField has this example in the plane:
TransformedField[ "Cartesian" -> "Polar", {x, y},
{x, y} -> {r, \[Theta]}] // Simplify
This is a general vector field:
TransformedField["Cartesian" -> "Spherical",
{f[x, y, z], g[x, y, z], h[x, y, z]},
{x, y, z} -> {r, \[Theta], \[CurlyPhi]}] // Simplify
