Why do DC motor controllers have minimum voltage?
The problem is not so much the H bridge itself as driving the H bridge.

The N mosfets are relatively easy. They are tied to ground so you can just feed logic signals into them directly.

The P mosfets are where it gets tricky. Their control signals must be relative to the motor supply voltage. To turn them off the control signal must be close to or greater than the motor supply voltage while to turn them on the control signal must be significantly less than the motor supply voltage.

So the control signal you need for the P mosfets is heavily dependent on the motor supply voltage. 

If the motor supply voltage is very low then you need a Negative gate drive signal to properly turn on the P-Fets.

If the motor supply voltage is sufficiently high to properly turn on the fets but less than or approximately equal to your logic voltage you can drive the P-Fets directly from your logic.

If the motor supply voltage is significantly higher than your logic voltage then you will need a signal of significatly higher voltage than your normal logic supply to properly turn off the P-Fets.

Motor controller chips are generally designed to operate in the third region.