Back again ...
Bearing cones are fine! I polished them and realized that it was just dirt that was giving impression of being worn. It's just the one seat that doesn't look good. In a way or the other they pressed those seats inside, but I'm not going to try to press them out until I don't have a spare tone arm. Also the idler arm is fine now.
So this morning I wanted to make a sound test, I reassembled everything, put an old Audio Technica 110E in the shell and "off we go". I didn't have any alignment tool, so I just put the cartridge in the middle of the shell and tried: not bad for the first try

Unfortunately I immediately noticed a strange hum once the needle is touching the vinyl

Until the arm is lifted, even with the motor running, I was able to raise volume up to three quarters without hearing anything (after the three quarters until full power a very little noise was audible) so I would say that there aren't any earthing or grounding problems. But once the needle touches the first groove then you can distinctively hear a sort of hum. It's a sort of ... It is like an extra vibration going into the needle ... of course you can listen to it until no music is played. Once music starts I cannot hear it any more.
I changed cartridge and put in a Denon DL160. Really nice sound, clear and very clean, even the lower frequencies were fully present. BUT that noise was still there. Here on LH I found the stylus gauge for my arm so I aligned it quite well. Sound was even better now, but that problem is not disappearing!
Conclusions:
- it's not the cartridge;
- I don't think it is the alignment;
- it's not earthing or grounding.
I would say that it's definitively not an electrical problem. The cause is not magnetic or electromagnetic. This is what I think so far.
Mechanical problems can be given by:
- the platter;
- the tonearm;
- the motor;
- the idler wheel or arm.
The platter: it is soooo heavy and I put the rubber mat;
The motor: quite noiseless;
The idler wheel/arm: some doubts ... the wheel is not that silent, but the noise you can listen to, once the motor runs, is not really constant. What I hear lowering the needle on the LP is a constant noise!
The tone arm: same doubts as above, even if I cannot believe that it can be those cone bearings ... .
Any suggestions? or what kind of tests should I do in order to find out what it is?
Thank you so much in advance for your help