How am I supposed to reconcile thermal equilibrium of a system (the CMB) with temperature differences in the system?
This is the basic argument for invoking the inflation period in the Big Bang model.
The equilibrium reflected in the CMB is a snapshot at the time of the expansion of the universe when the photon decoupled, at about 380.000 years:
The problem you point out is even more serious, because black body radiation, (ignoring the tiny inhomogeneities,) is the same from all places of the observable universe, BUT at that time special relativity did not allow for homogenisation. Many regions were outside the light cone that would allow homogenizing thermodynamic processes.
The quantum mechanical hypothesis of inflation was proposed in order to homogenize in energy density the early universe, the inhomogeneities attributed to quantum mechanical uncertainties. This model needs quantization of gravity, and at the moment effective models are used.
The observed inhomogeneities are the seeds for the observed coagulation of matter into clusters of galaxies and galaxies, in the Big Bang model.