There is a simple and compelling argument for why the Thomas precession (TP) does not give rise to rotational inertial forces such as the centrifugal, Coriolis, or Euler forces, for hypothetical observers or systems that are undergoing TP. There can be no forces of rotational inertia in such reference frames because the rate if any of TP is entirely observer dependent. An inertial frame observer sees TP of a gyroscope only if the gyroscope is simultaneously translating and accelerating transversely (i.e., crosswise) to the relative translation. The amount of TP, that is, its angular velocity or rate of precession, is then readily calculated as proportional to the vector cross product of the gyroscope's relative translational velocity and acceleration. Thus, because the TP is calculated by the external, inertial frame, observer and depends entirely on the relative translation and acceleration, it is not uniquely defined from the point of view of an observer co-moving with the gyroscope. Other observers in inertial reference frames moving at different velocities relative to gyroscope observe a different rate of TP of the gyroscope. Indeed, from the point of view of an observer moving with the gyroscope, there is no TP of the gyroscope at all, since the relative translation and acceleration both vanish. Therefore the co-moving observer cannot experience rotational inertial effects such as the centrifugal or Coriolis pseudoforces.
It is a very short step from making this observation to understanding the origin of the magnetic force, and probably the strong and weak forces as well.