It makes more sense if you think of clip space as an honest-to-goodness 4D space, and about the 4D-space line connecting those two points. The inequalities you mentioned define a pyramidal hypervolume in this space. The line is clipped to this hypervolume.
It’s probably easier to visualise if you toss out y and z and draw a picture of (x, w) space.
I think I see what you mean. Combining what you say with the reading I did last night, it seems as though the procedure is to interpolate in all 4 coordinates, in 4d essentially, and the wc is used just for calculating outcodes.