How is this load able to perform any useful work other than heating?
Consider a voltage source with some internal resistance, in series with a load. It turns out that the voltage source produces a power equal to \$I^2R_\text{total}\$ (this can be seen by using P=IV and Ohm's law).


This first assumption is incorrect. This is only true for resistive loads.
When you add reactance (inductive or capacitive), there will be reactive power which is not dissipated / become Joule heat.