Insulating the shaft from the frame with non-conducting balls is probably not a good idea. It's much easier to insulate the frame from ground.
You can measure the voltages on the circuit FETs without the capacitor connected: there is no point in trying to make it oscillate before the amplifier is working.
Are you sure you have connected the FET leads correctly? While the D and S can usually be interchanged, exchanging the G with one will result in a useless transistor.
Beryllium is alloyed to softer metals like copper and gold in order to stiffen them without work hardening them.
In this case that would be to make the copper clips retain their springiness over time.
I should think the chance of releasing beryllium particles to be vanishingly small, unless you decide to use an angle grinder on them or something.
Before proceeding further, you need to find out what the DC voltages on the two-FET amplifier are.
The drain voltages you measured indicate total failure of the amplifier circuit. It wouldn't hurt to verify that the frame is insulated from circuit common with a simple ohmmeter check.
With only a few tenths of a volt between your measurments of "Drain" and "Source" voltages, I still suspect that you have mis-wired the device. Note that the Gate is not the center connection, but one end.
In that circuit, the gate must be negative with respect to both Source and Drain, to reverse-bias the gate-channel diode. You did not state the actual voltages at the two gates:
in this circuit working properly, the gates are connected to ground through relatively large resistors.
By the way, if you insulate the shaft from the rotor with plastic balls, won't the leaf springs make the connection again?
Those voltages look reasonable. If you disconnect the feedback network (tuning capacitor still removed, only R5 to R8 switched from the gate of Q1 to ground) and feed a small audio signal (say, 100 mV rms at 1 kHz) to the gate resistor, what do you see at the output? Sometimes you need to check subcircuits before connecting them all together.
Looks reasonable. For the next diagnostic, replace the variable capacitor with two equal fixed capacitors and see if you can make it oscillate.
1 nF = 1000 pF is a bit high, but I would think it would work. If you can find some 100 to 220 pF caps, they would be worth trying.
When the "oscillator" flat-lines, are the DC voltages the same as when the feedback is connected?
It's conceivable that the two batches have different pinouts, but that would be strange. Probably just defective parts.
OK--it looks like connecting the feedback has no effect whatsoever--I would carefully re-check the connections around the feedback network to the gate of Q1--something is missing.
The LDR circuit you show really requires a photoresistor, for two reasons:
1. The photoresistor (also a thermistor or light bulb) is a two-terminal symmetric device where I(-V) = -I(+V), not necessarily a linear function.
2. Photoresistors have a slow response to the light compared with phototransistors. This can be exploited as part of the control design.
All this does have a down side; the tuning becomes roughly exponential. In other words, the frequency does not vary evenly as you vary the tuning pot. Ideally we would need a pot with an anti-exponential taper (good luck). As a compromise I used a log pot backwards, so turning it anticlockwise increases the frequency...
Turning ccw to increase frequency is a reasonable method.
Cheap log-taper pots are not very accurate, so you need to carefully calibrate the dial.
Accurate volume controls are usually stepped attenuators (totally unsuitable for your application) or stepped voltage dividers (which will work as a stepped resistor), which you may find awkward ( typically 24 steps).