If the source of magnetism is spin, why is magnetism only associated with charged spin particles? When talking about the source of magnetism, I have been told that there are two answers to this, as listed in this question. 1) we can think about this as the spin of an electron giving rise to the magnetic field (a semi-classical picture of a quantum phenomenon) and 2) magnetism is due to moving charges. The latter doesn't explain much about the source, but rather the realm in which we see this. I have since realized that there are particles that have spin that are not charged, so I am wondering why, if spin is the source of magnetism, why specifically charged particles are associated with the magnetic force? What is going on at the fundamental level here regarding the interaction between charge and spin to create magnetism?
 A: First of all, you are right when you say that spin is the root source of magnetization in (I would conservatively say most of) all solid-state based magnets.
Second, the ultimate interaction that gives rise to "strong" magnetism (with reasonable Curie temperatures of tens or hundreds of degrees Kelvin) is Coulomb interaction between electrons. According to the classis theory of ferromagnetism due to Stoner (see: https://en.wikipedia.org/wiki/Stoner_criterion), when Coulomb repulsion between (charged!) electrons becomes strong enough (there is a specific (Stoner) criterion on how strong this "strong enough" is), the electrons decide that it is energetically more favorable to align their spins in one direction (at the expense of higher total kinetic energy) thus making use of Pauli exclusion principle (that makes sure that any two electrons with parallel spins don't meet in real space) to reduce the total potential energy of the electrons in a solid. The stronger is Coulomb repulsion, the more energy electrons gain by going ferromagnetic. Neutral fermions could not benefit from such scenario and hence would have to rely on some other interaction channels (such as e.g. direct magnetic dipole-dipole interaction) which are as a rule much much weaker than Coulomb interaction.
EDIT: If you go outside of conventional (atom-based) matter context, where Coulomb is the dominant interaction, and look at, for instance, nuclear matter, then you can have an analog of Stoner magnetism with nuclear forces. Then you can in principle have a magnetic state of electrically neutral objects. Arguably the most dramatic realization of such "neutral" scenario is the neutron star, where constituent neutrons, - electrically neutral particles, - develop a collective magnetization.
A: As far as we can tell - in that we've actually searched for them and not found them - there are no magnetic sources (i.e. magnetic charges): the only way to generate magnetic fields is by moving electric charge. Thus, in order for an elementary particle without magnetic charge to generate a magnetic field, it must be moving. This motion can either be translational motion, or else it is angular (rotational) motion, i.e. spin. (That quantum spin is not "classical" rotation is not relevant; all motions, including rotations, of microscopic entities are quantum, not classical, motions.)
Likewise, it follows that if you have motion but no charge, there cannot be a magnetic field generated, just as if you have charge but no motion, there is no magnetic field either.
