I have a couple of geared hubs, and they are quite interesting and they develop good torque at low power. I have one of each on a 2wd ebike that is featured in detail on another thread here on ES. Balancing a DD vs a Geared hub was interesting for sure.
The problem I have is on a commuter, I don't want plastic gears. They just wear out too often for my liking. I commuted with DD motors for years, had plenty of hill climbing power, and all I had to do was charge the batteries and fix the occasional flat tire and replace tires when the tread was gone. Never a problem with the motor. The gearmotors WILL wear out and require maintenance. That's just my preference. It is rather like an electric car vs a gas guzzler. Less maintenance. But gearmotors do a great job at putting out torque in a small motor. However the larger DD motor can take a LOT more power. But that's another story.
Now onto the motor topic. First of all, this is covered thoroughly on ES in many threads on controllers. There are folks here who have designed them and I have participated in those threads and learned a great deal, even being an EE - it is a specialized field of design. Look for them if you really want to understand this stuff. Prepare to be surprised, things are often not what you would have thought. But in the end it is all straightforward physics - electronics and magnetics.
In any case there is very little voltage loss through the wiring and controller from battery to motor. On a high powered rig there is more or when trying to climb steep hills at low speeds with marginal wiring, but it is usually small compared to the voltage. So what we have left is the duty cycle and the current. The ONLY thing the controller can do is adjust the duty cycle, which lets some energy from the battery into the controller, or not. I'm going to ignore commutation switching which is also going on, that just follows the windings around. And we'll talk about trapezoidal controllers because sinusoidal is another level. So we have three wires to the motor and the commutation selects which two get power. The third is disconnected, or used as a sense wire for sensorless operation, which we'll also ignore here. So the operator opens the throttle, and the controller begins switching current into those two motor wires. The current is the same in all the wires (battery to controller to motor), the controller is only a switch at this moment. But if you have the throttle only 10% open, the switch is only open 10% of the time. So 10% of the time the current flows from the battery and 90% of the time it cannot flow from the battery because the switch is open.
Now here is where things get interesting. The motor is an inductor. So it resists current changes and it has stored energy in the magnetic fields. So it continues to push the same current even though the battery switch is open. The controller effectively has diodes to allow this current to flow - motor to controller to motor. So we have 90% of the cycle flowing this circulating current while the battery flows nothing. Then the cycle ends and the switch in the controller closes and again battery current flows. This cycle is the controller's frequency and it's what you can sometimes hear, often about 10khz or 10 times per millisecond.
Note that the circulating current is not free energy, it is from the magnetic field in the motor, and it is declining through the cycle, but the cycle is chosen to be short compared to the time constant of this current. When the power is switched back on the current also increases with a similar time constant so it makes a little sawtooth waveform. The voltage is a square wave, the current is sort of a constant value which is related to the torque the motor is producing.
So step back a moment and see what's going on in the system. The voltage is pretty much constant throughout the system. The current from the battery is let's say 20 amps, but for 10% of the time only, so the battery only drains at 2 amps average. The motor current is 20 amps for the whole time. So the motor current is multiplied by 10 compared to battery current in this case. The actual numbers in real life will vary and depend on the real values of resistance, etc. But the physics is unavoidable - the motor and motor wires see a lot more current than the battery leads.
Now one thing we can add to this - the current rotates between the three wires, but only uses two at a time. So the third wire has a chance to cool off. So that reduces the average current in each wire to 2/3 of the motor current. But that's still more than the battery current in our example.
The heating in the wires is from current, not voltage. I squared R is the power loss in the wire. Squaring the I really makes the wire power loss dependent on the current and it rises quickly as the current increases (doubling the current causes four times the loss). Multiplying by R means that thin wires also multiply the power loss in the wires. Power loss in the wires causes two things. One is loss of power that should be in the motor (it reduces the voltage to the motor). The other is heating and physically damaging the wiring and connectors. To save money vendors can put in smaller wires and connectors (partly because it looks better and more copper is more costly), and then they set the controller software to keep the current low. You'll still see full power at middle speeds, but you won't see it at low speeds, which is where we are climbing hills..... Which is exactly what we seem to be seeing here.
Incidentally phase voltage and current are misnomers for motor voltage and current.