Drude assumed that collisions were due to electrons scattering off heavy ions. When the distance travelled between collisions (mean free path) was calculated, the result seemed to agree with that assumption, as the mean free path was approximately the same as the spacing between atoms, suggesting that they were the cause of collisions.
I found it interesting how although we now know the Drude model's predictions for the scattering time tau and the electron velocity v_0 (from equipartition theorem) to be incorrect, there is still a number of quantities that are independent of the scattering time that the Drude model can successfully predict.
I agree, it is interesting that the Drude model can use simple kinematics to predict basic properties, even if as you say the relaxation time is not neccesarily justified as completely as the mean free path approach.
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