Executive summary: "Collapse of the wave function" is inherently supraluminal.
I suggest an easier thought experiment to demonstrate the apparently supraluminal (or FTL) aspect of a quantum mechanical measurement, without spin and polarizers.
A photon is sent through a beam splitter. The photon traverses both arms of the splitter, and is "everywhere" (or "nowhere") until it is detected at one end or the other. According to quantum mechanics, there are no hidden variables.
The two arms of the splitter extend a significant distance (hundreds of meters) in opposite directions. A single photon is injected at a specified time. A short time later the photon is detected at one or the other of the two detectors, labeled A and B, each a random half of the trials.
Clearly the two detectors do not have time to confer on the outcome. Suppose the photon was detected at A. How did detector B instantly "know" that the photon must not appear there?