Auto shutoff for LiPo battery

which sets the voltage on the batteryControlPin to LOW, removing power from the Arduino.

The above could be extended with a batteryLow LED that glows when battery voltage is below say 9.8 V.

When you remove power from the Arduino, what happens to those LED indicators?

I think rather than removing power from the Arduino, you want the Arduino to go into some emergency / low-power mode, like removing power to a motor, or reducing power to it gradually (emergency landing for example), etc. The arduino consumes very little power and removing power from it provides few benefits to justify.