...Current is measured in Amperes, and is something that is drawn by the load, not pushed by the battery...
I know that this is how it is commonly explained, though I'm not entirely sure why. (Perhaps to make it easier for people to predict how motors and other similarly less easily understood components with varying resistances will function in a circuit.)
At the more fundamental level, the voltage of the battery pack is what pushes/pulls the current through the circuit. Without voltage there is no current, regardless of whatever the "load" is. How much current there is depends on the resistance of the circuit and the voltage of the power source.
This is essentially the relationship that Ohm's law represents.