I can't tell the difference in your ontological alternatives,

whether we can consider the spin as truly pointing in the 𝑧-direction, or whether we can only speak of the 𝑧-component of the spin as being up.

Your spinor (1,0) may be thought to point to the north pole, if that helps you (but why should it?). If you started with a normalized version of (1,1), you'd see that a $\pi/2$ rotation of your detector around y would align the detector with x, and identify your state with the up x-axis eigenstate, so you might consider your spinor to be pointing to the +x direction. But why?

The Bloch sphere of your state tells you how to identify the "direction" of the spinor for any spinor.

I can't tell the difference in your ontological alternatives,

whether we can consider the spin as truly pointing in the 𝑧-direction, or whether we can only speak of the 𝑧-component of the spin as being up.

In physics, the two are synonyms.

Your spinor (1,0) may be thought to point to the north pole, if that helps you (but why should it?). 

If you started with a normalized version of (1,1), you'd see that a $\pi/2$ rotation of your detector around y would align the detector with x, and identify your state with the up x-axis eigenstate, so you might consider your spinor to be pointing to the +x direction. But why?

The Bloch sphere of your state tells you how to identify the "direction" of the spinor for any spinor.