Many questions have been asked on this site about the Stern-Gerlach experiment, but as far as I can tell this one hasn't. Does the following classical explanation of the SG experiment work?
Model electrons as a finite-sized hollow sphere of rotating charge $e$. Immediately after entering the SG device, the dipole moment of the electron quickly reorients itself to align with the magnetic field of the device. This is what I would expect to happen if a small bar magnet were placed in a region of high magnetic field gradient. Suppose this happens before the electron has traveled 1% of the distance of the device. Then, for the remaining 99% of the distance in the SG device, the electron is completely oriented either "spin-up" or "spin-down," so the binary "all-or-nothing" measurement is naturally predicted by this classical picture.
This would also work to explain sending the beam through multiple differently oriented SG devices, since the previous spin orientation of the electron is completely altered (very quickly) each time the electron enters a new device.
I don't think I was clear enough about the overall point of the question, since a few people have now brought up the fact that the spherical electron model has issues. I'm aware of that, but not really concerned with the specific electron structure model, so much as why some classical model wouldn't work to describe the binary output beam behavior (which knzhou gave a very nice answer to).
Perhaps a better hypothetical classical model would be: A classical point-particle with intrinsic angular momentum / magnetic dipole moment, the correct gyromagnetic ratio (supposing this could be a tunable classical parameter for point-particles), and some "braking mechanism" that allows it to quickly align with a magnetic field and stay aligned.
I think this question matters because the SG experiment is often used as a pedagogical example of quantum mechanics in introductory courses. When I first learned QM, I remember being confused about why this example was supposed to be so convincing, since it seemed that there could exist classical explanations of it.