For me, the Boltzmann distribution is caused by the dynamic motion of the atoms, not that the dynamic motion of the atoms is caused by the Boltzmann distribution.
Caused is a dangerous word here. Rather the Boltzmann distribution tells you statistical properties of the motion. Therefore, nothing is wrong with deriving results about the statistics of the motion from the Boltzmann distribution: The Brownian particles will be in thermal equilibrium with the fluid, therefore they will follow a Boltzmann distribution and from that we get the above relation.
A more microscopical method to get the relations for Brownian motion are Langevin equations (statistical differential equations). There you model the influence of the thermal bath by a stochastic force. In this model you can observe how the systems reaches thermal equilibrium from some initial conditions. The relation in question is reached in the formalism by assuming a time scale large compared to the damping time (which is equivalent to the assumption of thermal equilibrium). Langevin's original derivation also assumed the equilibrium by using the virial theorem in the form $\frac 1 2 m\langle \ddot x \rangle^2 = \frac 1 2 \langle T \rangle$.
The bottom line is: The Einstein relation is a result about the kinematics of Brownian particles in equilibrium, therefore it is necessary to require the particles be in equilibrium to derive it.