Can small wire gauge restrict current? powering a brushless motor
There are two separate issues. Will the wire get too hot? That is purely a matter of current and the wire diameter. The wires do not heat up instantly. Brief overloads will not cause the wire temperature to rise much, and the wire may cool down pretty quickly after the current stops. This is what happens with your starter motor in your car.

The second issue is, will the wire prevent power being transferred to the load? This depends on current, system voltage, wire diameter, and wire length. A 1V voltage drop in a 120V system is not a big deal. But a 1V Voltage drop in a 3.3V system is probably catastrophic. Note that wire length is equally important to wire diameter when considering power transmission. Short wires will still allow power to get through, even if they are thin.

For all these reasons, low-voltage, high-current applications (such as quadcopters and such) often need to use surprisingly large diameter wires. People also tend to underestimate how large wire should be when connecting a 12V inverter to a 12V battery.