The "linearization" of either the Klein-Gordon or the Schroedinger equation to obtain the Dirac or Pauli equation, respectively, is only a handwaving argument, leading to the correct description of particles with spin (in that cases spin 1/2) by chance.
A more convincing argument is the group-theoretical method used to systematically derive the single-particle observables from the (continuous) symmetries of space-time. For the non-relativistic case, the symmetry group of space-time is the full group of inhomogeneous Galileo transformations, which are decomposed as temporal and spatial translations, spatial rotations, and boosts, reflecting homogeneity of time and space, isotropy of space, and the principle of inertia, which states that the physical laws do not change for observers that are in uniform motion with respect to each other.
The next step is to analyze, how these symmetries are realized in quantum theory. First of all one considers one single symmetry transformation. As has been proven by Wigner (and later simplified by Bargmann), such a symmetry transformation can be represented on the Hilbert-space vectors as either a unitary or an antiunitary transformation. If one has a transformation that is continuously deformable to the identity the transformation must be unitary, and since we consider only transformations which are continuously connected to the identity, we have to look for unitary representations of the Galileo group.
Now, there's one subtlety in this. In fact the (pure) states are not really represented by the Hilbert-space vectors, but only by these vectors modulo an arbitrary phase factor. That means that one needs not have unitary representations but only unitary ray representations, which are representations up to phase factors.
This has two very important consequences for physics: First of all the most general transformation is not necessarily the classical Galilei group but its covering group. That means that we are allowed to use the SU(2) to represent the rotations (making the group, SO(3)) within the Galilei group.
Second the Galilei group is such that it admits the introduction of a socalled nontrivial central charge, which is an observable that commutes with the generators of the one-parameter subgroups of the Galilei groups. The latter make the energy (Hamiltonian) and momentum (generating temporal and spatial translations), angular momentum (rotations), and boosts (center-of-mass position). The central charge turns out to be the mass of a particle, if the irreducible ray representations are interepreted as defining elementary (non-relavistic) particles. As it turns out, the representation without central charge, i.e., particles of zero mass doesn't give physically meaningful representations of the Galilei group (a famous paper by Wigner and Inönü).
The physically meaningful representations of the quantum-Galilei group, lead to representations for a particle, which has two intrinsic quantum numbers, namely its mass m \in \mathbb{R}_{>0} and its spin s \in \{0,1/2,1,3/2,\ldots\}. The spin determines the behavior of the one-particle state for particles at rest (zero momentum), i.e., for s=1/2, the zero-momentum states span a two-dimensional spinor space. Since in non-relstivistic physics, the spin commutes with momentum as well as with position operators, one can build a basis as either the direct product of momenum-eigenstates and spin-eigenstates or of position-eigenstates and spin-eigenstates.
In terms of wave functions this leads to spinor-valued wavefunctions \psi_{\sigma}(t,\vec{x}) or \tilde{\psi}_{\sigma}(t,\vec{p}).
The Pauli equation, including the correct gyrofactor of 2 (!), can be derived by using a specific form of minimal substitution to couple the electromagnetic field to the matter field in such a way as to make the invariance of quantum theory under changes of the wave function by a phase factor local. Thus, indeed one doesn't need relativity to make sense out of the spin observable, and also the gyro factor of 2 pops out of gauge invariance and the principle of minimal substitution rather than relativistic (Poincare) covariance.
However, in relativity the same analysis of the Poincare group as done above for the Galilei group (with the important difference that for the Poincare group there don't exist non-trivial central charges, and mass is thus not a central charge but a Casimir operator of the Poincare group) leads to the possibility of zero-mass particles, and if one considers massless particles with spin 1, these must be gauge fields, if one doesn't want continuous intrinsic quantum numbers, which has never been observed to exist. Thus, relativity leads necessarily to the principle of (Abelian) gauge invariance.
)
