on battery terminals causes electrons to flow naturally through a circuit. (Like an object in a gravitational field.)
Probably better to think of it as pressure in pipes with no gravity, since circuits are loops.
(The reservoir is not needed, though, and can be removed from the diagram without changing how it works.)
I understand that electrons will flow through a circuit at a constant rate and not be consumed by the circuit. I understand that these particles have higher energy before being applied to a load and lower energy after being applied to a load.
I don't understand what is doing the work here, since the electrons are not being consumed.
Think of the electrons like links in a chain (except pushing each other instead of pulling). The chain is not consumed, it circulates around forever, but it can carry energy from one wheel to another. As you turn a crank on one end, energy is transferred by the chain to the other end, where it can do work.
Does movement itself work on the load, like friction?
In a resistor, yes. The resistor converts electrical energy to heat energy through "friction", in the same way friction converts mechanical energy into heat energy. (Imagine that your chain is replaced by a belt, and the belt is rubbing against something at the other end to produce heat as you crank the local end.)
Technically, it's complicated and requires quantum mechanics to explain, but to oversimplify: Impurities in the metal crystals and vibrations from their temperature cause the electrons to collide with the metal and transfer energy to it. (That's why resistance increases with temperature, for instance.) But it behaves like friction. See How do you explain electrical resistance?