Current in a closed DC circuit
However if I connect the two terminals of a battery together with a simple wire then voltage will flow until the battery is dead. So if there is voltage flowing through the circuit why is there no current?


Current will flow in this scenario. Who told you it won't? 

Voltage doesn't flow in this circuit or any other.

As an analogy, say we have a river whose current flows from a high altitude to a low altitude. Altitude is analogous to voltage in this situation as it determines how quickly the river current flows. But the altitude doesn't flow in the river.


  there is no measurable current in a closed DC circuit when there is no power consumption.


This isn't a general rule for all circuits. If a wire has very low (approximately zero) resistance, then current can flow through it with approximately zero power consumption. 

Only in circuits with non-zero resistance would it be true.