Thanks all for your concern.
Once I realized, as mrkev said, that the aux was shorted, I disconnected the cap, as indeed, this was much worse than with the old cap that had a high resistor value.
Unfortunately, in my tests the motor heated a lot, and maybe now it is worse than in the original state.
I could start it manually, but it would stop, and now does not want to start again.
So now I understand that the only way would be to rewind it, but it is too much work. I will just leave it like that,
and provide to my friend my own grinder which is a cheap chinese one on which he could put his stone.
And if I manage some day to find a electric motor of the same kind, I will replace it,
as I find the casing very nice.
So it is an example of total failure in searching to fix something, but at least I learned something.
Thanks to all for your contributions !