Have you looked at Drude's model? I was taught something like that back at school and have kept it in mind as a intuitive way of understanding it.
We want to understand why the current (rate of flow of charge) should be linear with the potential difference.
The Drude idea is, as you noted, related to friction.
Firstly, the EM field is linear in the potential difference. This generates a force on the electrons in the conductor.
The electrons then accelerate with the field. Were their path unimpeded they would constantly accelerate. Instead, they 'collide' with atoms in the structure of the conductor and 'bounce' off. In steady state, the rate of specific impulse transferred from bounces has to balance the force due to the EM field.
The key observation by Drude is that the velocity of the electron at the time of collision will be directly proportional to the strength of the field.
Please be clear that this is only an intuitive model, albeit one that gives surprising insight despite being fairly crude.