By no means not a complete answer, more a criticism of @luksen’s one. It is posted here because the text is too long to fit in the comment field.
First of all, the spin is not a well-defined concept for composite particles. More precisely, whether the spin of a particle is defined depends on how the “particle” is defined. Look at an atom: it has the nucleus and electron shells. In many cases both have non-zero angular momentum, that implies they are coupled. There are several slimly different energy levels, and if you look at each of them as at a particle species, then yes, each has its spin.
Do such things as the Stern–Gerlach experiment capture such “particle species”? They don’t, because magnetic moment of electrons is strong (induced both by spin and orbital angular momentum), whereas the one of the nucleus is weak. Electrons vigorously interact with the external magnetic field, while nuclei reluctantly interact with either one. The species which are captured by a Stern–Gerlach-type device are superposed states of these slimly different energy levels, not spin states of a particle with a well-defined spin. By the way, for any silver atom of stable 107Ag and 109Ag isotopes a trivial calculation shows that it must be a boson as a whole and can’t be a fermion; this thing is downplayed in most QM texts.
What did the Stern–Gerlach experiment actually demonstrate, indeed? Electrons’ spin and their orbital angular momentum (more precisely, of the nucleus–electrons system) play all the drama. The nuclear spin was almost invisible because of much weaker magnetic moment. The experiment exposed electron shells and concealed nuclear spins – that’s why an impression of a fermion. I stress it: not an actual fermion. Only such apparition. No particle the spin is measured of.
So, why the apparition of spin ½ when the silver possesses 47 electrons? In fact, because of electronic configuration. Only the 5s electron contributed since the remainder forms completely filled subshells. What is the 5s electron? Since it is an
s (ℓ = 0) orbital, only the electron’s spin matters. But it is only the silver. What happens to incomplete subshells in other cases? For d1 (it is the case for scandium, yttrium, lanthanum, lutetium) we have contributions of both electron’s spin and orbital angular momentum (ℓ = 2) to the magnetic moment; and there are several metals with more than one incomplete subshell. What will happen to other types of composite particles, such as numerous flavours of exotic atoms? I won’t predict. Finally, what remained of the fairy tale that spin and angular momentum are the same thing? Not so much.