The load is not the one doing the work. How does the load draw from a dead battery? Your linguistic desire is equivalent to saying that the force of the motor does not propel the car, instead it is the mass of the car which draws the acceleration from the motor.
As I've already stated, it's clearly Ohm's law, which describes the relationship between voltage, resistance, and current flowing through the circuit.
As it's the resistance(ohms) presented to the source's potential (voltage) presented by the load, and not the capacity of the source, that determines the current flow(amperage) in the circuit, the accepted way to state this is that the load
draws current.
Note the title of this discussion:
How to calculate current and voltage draw of a single resistor circuit?
A quick google search for "current draw" and looking for the phrase on the pages reveals that institutions like the IEEE, NEC, NIST, electrical product catalogs, power calculators, engineering forums,etc.. all use that terminology.
So you can object and disagree with my posts if you like. But if you want to use terms that are considered correct by professionals and standards bodies alike, as well as communicate the concepts to others accurately, then you are barking up the wrong tree.