quantum mechanics - fixed input qubit state to an arbirary pure state using two variable rotations and one fixed rotation

It is a theorem that any arbitrary unitary transformation in SU(2) can be factored into the following form:

$ O = U_X(\theta) U_Y(\phi) U_X(\delta) $

Where $U_X$ is a Bloch sphere rotation. I believe it is possible to fix one of these angles, leaving only two angles as degrees of freedom. So, I am saying that, given that X and Y might be different, and the fixed angle is some number like $\frac{\pi}{2}$ any unitary can be written in a form something like:

$ O = U_X(\theta) U_Y(\frac{\pi}{2}) U_X(\delta) $

Does this sound right? Perhaps I am working on the assumption that, if a pure input state is fixed, then you only need two degrees of freedom to map that input state to any other pure output state. This thinking comes from the fact that the pure states are all on the surface of the Bloch sphere.

Here is a quote from Wikipedia:

Since polarization states are defined by two degrees of freedom, for example azimuth angle and ellipticity angle of the polarization state, such a polarization controller needs two degrees of freedom. The same holds for the task of transforming an arbitrary polarization into a fixed, known one.