For detection, yes, it can be done by an extremely simple circuit. But you would need the raw I/Q sample data over a broad frequency range to perform a technical analysis to identify the nature of the microwave. Here's when a SDR comes handy.
The LED just reveals power (detection of radiation over many visible orders of magnitude) ... it doesn't need to detect coherent I/Q, because I don't think neural effects are sensitive to phase either. Selectively cooking a brain would require relatively high power and beam forming, but not high data rate. Audio frequencies are visible (use multiple band pass filters if you like) .
You'd also want a controlled test source so you'd know if the attacker had burned it out.
With the Schottky you will get better low power detection (lower forward voltage 0.18V vs 1.8V), but it doesn't inherently indicate activation (you'd want an electret or piezo speaker at least) with the same dynamic range as your eye is capable of with an LED. Of course if you have an RMS power meter with uW resolution it will be OK... but LEDs are really good.