Here is an experiment of an LDR used as an amplifier.
It uses a neon tube, but a common LED can be used, improving efficiency possibly.
I have done this using a LED and it amplifies.
The drawback is the slow response of the LDR, so it is suitable only for low end audio ranges.
However, very big voltage amplification should be expected, as a low energy light source could vary "modulate" a possibly high voltage.
I read "has not been fully developed" as "I did never test the circuit but believe it could work".
Does it actually? I would expect the glow discharge lamp to fall into relaxation oscillations when shorted by a capacitor in series with a low impedance audio amplifier output. If the circuit actually provides power gain is a different question.
As hinted in the article, it works to amplify 60Hz. What it doesn't say is it's gain drops rapidly as frequency increases. It doesn't have any power gain, it just converts input current to an output voltage and only when the impedances are very high.
You can see the same effect with an ordinary optocoupler and it will work at much higher frequencies and lower voltages. Just use a high value load resistor on the optocoupler output (~47K with 5V supply) and set the LED current so it gives about half supply volate across the load. If you apply a modulating signal to the LED you will see a larger change in the voltage across the load. The gain is very limited and it will not be very linear but it will work. As it costs more and gives far poorer performance than a single transistor, it has rarely if ever been adopted commercially.
You can get similar amplification with some voltage regulators. Modulate the 'Adj' pin and load the output pin.