Bypassing the cathode of a triode increases the gain about 6dB, but only down to a certain frequency. Increasing the capacitance lowers the frequency before the gain drops off.
The phase shift oscillator works by feeding the signal from the plate back to the grid through an RC network that shifts the phase 180 degrees. The RC network must not attenuate the signal amplitude too much or the tube will not oscillate. The tube needs to have enough gain so that together with the amount of attenuation of the RC network, enough signal gets back to the grid to cause oscillation. Note that the RC network only has 180 degrees of phase shift at one frequency. Above that frequency, there is less phase shift, below that frequency, there is more phase shift but also less signal that gets back to the grid.
So two things can cause the oscillator to stop oscillating. The gain of the tube being too low and too much attenuation in the RC network at the frequency where the phase shift is 180 degrees. The attenuation of the RC network changes as you adjust the 3 meg pot. The best way to evaluate the RC network is with a circuit modeling program because every capacitor (3) raises the order of the equasions. Generally try to keep all the capacitors about the same value.
One other thing that should be mentioned, that is the B+. In a tremolo amp such as this, there is quite a bit of the tremolo signal that gets on the B+ because the current of the output stage is changing. This acts against the oscillator like negative feedback. A symptom would be that oscillation stops or changes frequency when you turn up the Intensity control.
Tubes manufactured these days don't have quite as much gain as in previous decades. The one you have just might be low gain, try another brand. A low heater voltage will also cause the gain to be low. Check the heater voltage at the socket and check the socket for tight contact to the tube pins.