I wonder if there's some simple variable transmission that could be electronically controlled to deliver power from the motor to the propellers in highly controlled fashion.

A generator and four motors. Efficiency would probably be around 70%. It's about 90% in diesel-electric locomotives where more weight and money is spent on efficient conversion hardware.

That adds weight of 4 beefy electric motors to the system.

There's differential based continous variable transmission that varies according to resistive torque on a control shaft: https://makezine.com/2012/08/10/lego-continuously-variable-t...

This might enable electronic control witout using any motors just one simple coil per propeller. Energy generated through this way of control could be stored to power onboard devices.