Spin is not an object really 'spinning', but in fact, neither is angular momentum, and momentum isn't really an object 'moving'. Let's be clear about what momentum is in quantum mechanics: there is, say, an electron field, and along a particular path it oscillates as `e^i(Et-px)/hbar`. The momentum determines how frequently the wave oscillates as you move in space; the energy determines how much it oscillates in time (the E and p operators pull E and p scalars out of this, and the Schrodinger equation says that E^2 - p^2 = m^2 (sorta; we take the positive solution of a low-momentum expansion...).
Anyway, the point is, momentum means "the wave function oscillates as you change the position'. Angular momentum means "the wave function oscillates as you change the angle'. The base orbital angular momentum state looks like `e^i m φ`, and the operator that extracts `m` is `- i d_φ`. Etc.
The other thing about wave functions is that they are continuous everywhere. The presence of a particle is something like "having a knot" in the field -- there is a region where there must be some net object, because if you integrate around the region you see non-zero net flux. That kind of thing. So to have "intrinsic angular momentum of 1/2" means that, if you integrate around a region where a fermion is, you'll see a net rotation of the wave function by half a phase.
Now, that seems nonsensical, because if you integrate around a point, you should get back to the value you started at. And in fact, you do, but the two are distinguishable: the reconciliation for this is related to the fact that SO(3) is not simply-connected; if you produce a path of rotations that takes every vector back to where it started (such as XYZXYZ, where X rotates around the X axis), any path that performs one loop does not deform the identity path, but one that does two does -- which makes these states physically different. So if you gave me a wave function, I could bucketize all of the points in, say, the 'z' direction into ones that are in the 'identity' element (relative to a point of my choosing) vs the 'anti-identity' element. These have opposite spins.
I am still working on tightening up the model, and I haven't quite figured out how this causes the magnetic field to send such a particle in a different direction. But the rest feels close, to me, and doesn't rely on any hand-waving statements like "because it seems to work this way".