First, to avoid word confusion, let's speak about arduino board one one side and open theremin shield on the other:
The arduino board is just hosting the Atmega ship and all needed stuff to have it working (PSU, clock, connectors, serial port, USB, ...).
The volume oscillator is on the open theremin schield.
TP-6 and TP-7 are test points. You normaly don't need to solder anything there. You may use it to put the oscilloscope probe and track the signal.
Even without antenna the oscillator should oscillate. So it's probably not a bad contact between antenna and open theremin Shield.
This lead me to an essential question: do you have access to an oscilloscope ?
If yes you can track the signal: Do you see 460 Khz at TP7 ? (oscillator working ? )
Do you see it on arduino board at "Pin 4" ? (continuity ?)
If you don't have an oscilloscope, you still can track continuity with a multimeter, track power supply at U5 4069 ship, ...
But you won't know if your volume oscillator works correctly.
The arduino being sandwiched with the shield won't help investigations, of course. some "jumper wires" (search "arduino jumper wire") may be usefull.
Continuity: solder joint are one thing, you may also search for broken tracks (less probable but we never know...), ...