basic experiments

Oh yes, you need a lot of spare time to find that out, which I'm afraid neither of us have.. Is there a professor out there looking for a physical measurement internship for students, a thesis?

I have tried some simulations in LTSpice. Also time consuming, but the program does most of it autonomously. Here a first result. I analyzed the time signal of identical two mixed oscillators with a buffer to decouple them and vary the C (hand capacitance) of one. Coupling is made with k factor of the coils. Every point in the diagram is a single simulation and estimation of the beat frequency after demodulation. More points and values can be inserted next time.  

Without coupling you have the strong linear proportionality of hand capacitance to the pitch (math please find in this thread some pages before). The coupling compresses the pitch behavior compared to the hand capacitance and the pitch abruptly collapses to zero.

Don't confuse this with the linearity of the pitch in relation to the hand distance! The red and the green values gives better pitch linearity versus hand distance. .

"Coupling is made with k factor of the coils."  - JPascal

I'd imagine the coupling on something like the Etherwave, where it happens via capacitors going to a non-linear mixer, and the pick-off point is between the tank an the EQ coils for the variable pitch oscillator, must be a rather complex thing?  Still definitely worth pursuing this, and simulation is likely the best initial route!

The known waveforms in the lower audio range of the Etherwave thermemins are asymmetrical within a time period. The lack of a buffer leads to capacitive coupling between the fixed and the variable oscillator. 

Simulations in LTspice can show similar results to these typical waveforms when using simple oscillators with a LC circuit. Here too, the effect of a zone with zero beat frequency occurs depending on the coupling.

The offset correction must be taken into account. There appears to be compensation for the increased coupling. 

The values for simulations are:  oscillator_1 L = 4 mH; C = 15 pF, oscillator_2 L = 4 mH; C = 15 pF; C_hand = 0.02pF; C_hand = 0.013pF
 A) green 100 Hz, red 55 Hz; coupling 2x Cc = 0,30 pF
 B) green 90 Hz, red 33 Hz; coupling 2x Cc = 0,35 pF
 C) green 104 Hz, red 53 Hz; coupling 2x Cc = 0,35 pF; offset correction due to oscillator1 C = 14.998 pF  

Now some theoretical considerations. The coupling can be described as an exchange process between the two oscillators. In NMR, for example, this is used for dipole-dipole interactions of spins. The degree of coupling leads to a broadening of the resonance curves, their maxima converge and finally collapse to a mean resonance. The coupling is described here with a time constant whose reciprocal value is a measure of the bandwidth.  And yes, the critical coupling of bandpass filters can also be represented in this way.

Here you can find some results and formulas for this effect. If you find a fundamental or computational error, feel free to post it here.  It has been a too long time since I have dealt with such mathematical questions.
            
The first diagram shows the frequency behavior at a certain coupling constant. Without coupling, very narrow lines would result. The difference between the maxima is the beat frequency. With coupling, the beat frequency is lower.  The second diagram shows the beat zero zone with different couplings when only one maximum is reached. It looks similar to the solution proposed and implemented by Ilya for his theremin simulator program.