single-capacitor oscillator choice for digital theremin
I think your best bet is the good old colpitts oscillator - nothing ends up in saturation (unlike a square wave oscillator) and with even modest transistors, it's capable of several tens of MHz: -



In a slightly more advanced version it is used in capacitance probe measurement circuits looking for pieces of metal rotating past the probe end and, it will certainly detect tiny fractions of a pico farad change.

Attach the antenna (aka probe) at the top of L1 or even make two flat plates (one being 0V) where you can wave your hand over and change the frequency. To make it more sensitive to habd movement, the two 1n0 capacitors should be made lower in value and L1 increased.

It produces a sine wave but this can be changed to a square wave using a fast comparator like a MAX999. The basic oscillator will also run down to 3V (just about) on the power lines.