I got back to my vibrato issue as I had few days off. I have spent quite some time to know more about the cause and eventually find the cure. To this day, the issue is not solved but bypassed. It is OK for me "as is". This is what I have done, just for the record.
I took the preamp out from the piano and I built a separate +/- 15 V lab power supply to see if I could reproduce the problem without the Rhodes power supply. Which I did. The Rhodes power supply is cleared.
I changed all the electrolytic caps which were all faulty. I also changed a non-polarised cap on the oscillator (C8). I removed the vibrato switch and put a shunt in place. I cleaned the board carefully. With no better result.
One interesting thing to notice : the voltage between the -15V rail and ground drops to few volts when the output of the oscillator (op-amp pin 7) is at low state. This is why the pilote indicator led (LED 4) gets dim. And this is why the sound gets distorted. The signal is clipping. This can be observed only when the oscillator is at a low rate as the the voltage has enough time to drop.
I have the feeling that the op-amp output at pin 7 is short-circuited to ground when at low state, draining the power supply. My guess was that the vibrato indication led or its transistor was the culprit. Disconnecting them from the oscillator did the trick indeed. However I did my best to find a failure there, without success. I even replaced the transistor to a new one... it did not do the trick.
I ended up leaving the vibrato indication disconnected from the oscillator output and left the led on : I removed R26 and the transistor. I added a jumper from R28 to the led.
Back to the piano, the preamp and oscillator work fine now. Without vibrato indication however.
I also connected the third pole of the switch to ground.
Picture here :
https://photos.app.goo.gl/Srw82URnJhFiVtXr9