Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:
As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.
Leads to the Klein-Gordon equation.
Leads to the Dirac equation.
Leads to the Proca equation.
Theorized for the graviton.
More interestingly, how is that implied by the Stern-Gerlach experiment? suggests that half could either mean:

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