Current (measured in amperes) is what powers a motor by inducing magnetic fields, but voltage is required to get the current flowing. Brushless motors (like the familiar hub motors) have a Kv rating giving the motors natural speed in RPM/volt. Technically, the Kv rating specifies a line in the speed-voltage space along which the "back emf" (voltage generated internally by the spinning motor) equals the driving voltage.
The controller is constantly turning the power on and off at high speed, providing an average voltage to drive the motor, and the current through each coil corresponds to that voltage. Increasing the throttle tells the controller to increase its duty cycle (ratio of "on" to "off" time), providing more power to the motor. (for brushless motors, the controller does this for each of three phases and also maintains motor timing.)
Overheating is the biggest risk from running at high power by increasing voltage. Heat can ruin magnets, melt insulation, burn lubricants, and change mechanical tolerances in a motor. In theory, a controller could limit current at lower speeds, limiting total power regardless of its input voltage, yet still allow that full voltage input at high speeds when the motor is spinning too fast to draw peak power. Stepper motor controllers operate this way, but I don't know about the features of available brushless controllers.
Lastly, I agree - you'll need a motor with a higher no load speed higher than you plan to travel. Power will start to drop quickly as you approach the no load speed. Your best bet is to ask other riders about their experience with combinations similar to the ones you are considering.