I have some news.
I unmounted the steam temp probe. It looked just brand new. I probed the connector and measured a steady 65k resistance. Then I asked someone in the house to keep the probe in the palm of his hand while probing with the multimeter and the resistance went slowly down to 50k. This makes me conclude that the temperature probe is not the issue unfortunately. I cleaned and brushed the probe anyway, mounted it back and confirmed that the issue is not fixed with a test and steam pressure rising up above 3 bars.
During the test I unmounted the bracket where the SSR is sitting to have a clear view on the led.
Here is some pics taken with the steam pressure at 0.5, then 1.5 and finally 2.5. The red light goes on as I turn on the main power switch. I do measure 12V DC on one side and 12V AC on the other.
I am surprised anyway by not seing this led turning off to indicate that the heating element has been cut. On a Lelit video here we can clearly see how the relay is supposed to work:
The SSR in the video is a little bit different but the green light turns on around 5 seconds after pushing the power button. The SSR red light on my machine immediately turns on on power and stays on all the time which seems a bit suspicious.
Coming back to the temp probe, I do not have the light fault combinations to indicate that the probe is faulty or disconnected so I consider this item as good and working.
I decided to disconnect the SSR Relay and probe it out of circuit. On the 12v DC side I measure OL so open circuit and on the 12V AC side I have continuity. This is odd as a good relay should measure Megaohms or no continuity on the AC side when not powered. It is like this relay is stuck open. I may have the culprit here :-).
I may at this stage try my luck and order a replacement relay?