The LFO is on the edge of working. The power up transient kick-starts the LFO, but it doesn't have enough gain to sustain oscillations. A common problem with phase-shift sinewave oscillators.
I see you used a solder-lug pot for RATE. Are you sure it's B100K? If the two gangs don't track well, that could also stop the oscillator. It can be checked in-circuit. With the power off, set RATE to 0. Measure resistance from ground to pin 3 on one gang and then the other. You have to wait for the reading on the meter to stabilize. Verify that the resistance values match within 10%. Then set RATE to noon and repeat.
Try removing all of the electrical tape on the RATE pot. It might be introducing a leakage path.