Theo presented the following explanation :
In this case the circuit is carefully designed to be sensitive to voltage.
A regular automotive indicator LED bulb is a 3.2V white LED and a series
resistor which drops about 10V. The resistor limits the current so the
forward voltage of the LED is immaterial. In my setup the current is very
much dependent on the forward voltage, which is the point.

I'm not sure I'd be able to find LEDs when put in series with quite the
right voltage/current curve though. It might need a more conventional
cutoff circuit.

I doubt that will work, LED's dim appreciable as the voltage reduces,
so they have no clear cut off voltage. The only way to get it to work
is to have the LED's very dim at full voltage, too dim to be useful.