If the spindle can be rotated to any position ( with servo motor ) and locked, and the "X" axis can travel completely across the face of the work ( although only in the vertical center ) It occurs to me that ANY position on the face can be reached WITHOUT an "Y" axis.
It seems you want to do work in the Z axis direction, with a live tool pointing along the Z axis, no ?
You can spotface any features or spots on the face, yes.
You can drill, and bore, and tap features on the face if you have suitable tooling/sw/cam or handwrite the programs.
You can make a groove (slot, mill features) along the face, using a combination of C axis angular position, and positioning in x, theoretically, with feed movements in Z.
This is challenging in terms of maths.
I don´t know that cam sw supports such combinations, or which ones do.
Considerations.
The torque requirements are high on the outside edges, vs the center part, for live tooling.
The angular displacement of the C axis is not linear, to get a straight-line path.
It varies according to both the vertical distance in y, and the distance in x.
Basic pythagoran math can solve it, z pwr 2 = (x pwr 2 + y pwr2).
Chop the path into lines, solve all endpoints.
Then solve needed angular change for C, from one point to the next.
A macro will easily do it, in excel (word, any office applet) or any programming kit of your choice.
Write to file with gcodes as needed.
Use *variables* !
This allows using slower/fester feedrates in a somewhat intelligent manner without alwasy calculating the whole thing.
This allows smaller/bigger steps as needed, and testing/solving for constant speed / exact stop more.
Short answer
milling/ drilling with the tools orientated in the z axis no
centerline drilling in x orientation no
milling/drilling off centerline in the x axis you would need y axis