Physics is the discipline of studying nature and modelling it with mathematics, so that one can predict new observations and measurements.
In the process there are several frameworks where physics models exist, because of kinematic and other variables, and it can be shown mathematically that in the overlap region they merge or emerge from each other.
To have a mathematical model one has to impose extra axioms, called laws, principles,postulates in order to pick up those solutions that describe and predict data.
In the classical electrodynamics of Maxwell, it is not necessary for electric and magnetic fields to be related with motion in order to exist. There exist monopoles in the theory . The electric monopoles have been discovered in particle data, and it is the electron to start with (and quarks and muons ..) , but no magnetic monopole has been discovered (as yet?).
In classical theory it is not necessary to have a special relativity kinematics source for a magnetic field .
The world of particles is modeled with quantum mechanics, and within its axiomatic assumptions there is the table of particles with their charges and spins.
Axiomatic spin definiton for particles is necessary in order to have the law of angular momentum conservation at the particle level.
But I don't know how you would explain the magnetic field due to its internal magnetic dipole through Special Relativity because the charge is not moving at all.
Can you eliminate the electric field of the electron by special relativity motion?
The confusion comes by using classical electromagnetism where quantum modeling is needed. In the quantum frame , given a spin , nothing has to move, but it has been found that particle spin can be treated classically ( as there is continuity between classical and quantum) and since there is spin and mass, a magnetic moment a la classical EM can be calculated. Note that the classical calculation needs a correction factor.
It is useful to check the classical model behavior at the quantum level, it is not binding, the quantum is a different framework with different mathematics, and one has to be careful to keep to the correct framework when using mathematical tools from a larger one.