Best visualization of electric current and voltage
The Drude model is not intended for 'visualising' what is going on.

The Drude model is a very good (for the time it was introduced) classical way of putting some equations onto what was observed, to try to understand what was going on numerically. As such, it gives plausible resistance predictions and handles the the change with frequency for many metals, but fails badly for sodium. It also makes a reasonable fist of some specific heats, though due to compensating errors that are in the order of a factor of 100.

How do you do 'better' than the Drude model? You can make it more like reality by introducing quantum mechanics, for instance Drude-Sommerfeld and the free electron model. That is much more successful than Drude, and certainly does away with electrons 'hitting' atoms, but no-one in their right mind would argue that introducing quantum mechanics makes something more intuitive (for me, visualisation implies intuitive, or at least classical).

One alternative is to crank the realism down a notch or three, and use the hydraulic model, which laptop2d has already illustrated, albeit briefly. You can push the hydraulic model a long, long way before it breaks. Specifically you can have inductors, capacitors, batteries and generators (we even have a hydraulic switch mode power supply, a boost converter, the hydraulic ram!)

You can't have realism and simplicity. Try to intuit how 'particles' like electrons behave, and one is already lost. For instance, an electron is not a billiard-ball-like particle, even in a system as simple as the hydrogen atom. Add another, and they interfere in wave-like ways. Add more, and an electric field, to a lattice of ions, and pandemonium breaks loose.

The best we can do is to have a 'magic fluid' that obeys certain more or less plausible rules. Whether you prefer water in pipes, classical electrons hitting atoms, electrons in bands being scattered by the lattice, or just plain-old 'conventional current flow' with no attempt at visualisation, is up to your taste. It's just the way the world is.

What I do, and from conversation most other engineers do as well, is to stick with a magical conventional electric current, which is a bit like water, and take the 'plausible rules' on trust. Occasionally, we have a crisis of belief, and take a peep over the wall of Drude and then quantum mechanics, to see if what's going on over there looks plausible, decide it's too complicated but probably sound, and come straight back to hydraulics and circuit simulators. 

Unless you're the scientist designing the next semi-conductor material, so need to know about band-gaps, you can do electronic engineering never bothering about anything more than conventional current.