There are some advantages to running lower voltages. Batteries are easier to deal with for one.
If a motor is designed to run on, say 12v, it will have fewer turns of heavier wire. Iron saturation is a function of the amp-turns (amps times number of turns), not just the amps, so that works out to be equal for a given motor size.
The big problem is resistance losses in the system, which as pointed out above, increase with the current squared. Still, you can largely design around this. Fat wire costs more than skinny wire, but you don't need very much. Again, the motor has to be wound properly, but there will still be a disadvantage to running a lower voltage in terms of winding resistance. The controller is where you have the biggest disadvantage. If the controller is disigned for a lower voltage, it's easier to find FETs that have really low on resistance. This may be hard to find in the marketplace. I've seen a lot of big pallet jacks that run on 24v though.
So, if you use fat wire, and have a motor with fat wire, you can be almost even with a higher voltage system, but the controller and motor heating will be limiting factors at higher power.