The mystery was that one day to another it stared eating needles, and in some cases spat out the broken pieces at high velocity in all directions.
The good news is that you can spin a sewing machine very slowly by hand to see what's going on and while doing so I saw that - obviously - the needle was going down into the spool chamber at precisely the wrong moment, with the spool chamber still being closed. It's a very intricate mechanism that passes the needle thread under the spool using some black magic to pretend the mechanism comes apart while it does the move. Timing is super precise.
Like in the card reader example the origin of the problem was a part that had slipped on shaft. Apparently a bunch of thread had gotten stuck solid enough that it had wedged one of the cams hard enough that the shaft inside it had continued to move.
Even after finding the cause it still took forever to figure out the right setting (no markings on the shaft, no service manual for that particular machine indicating the right angle) but eventually I got it working and it still works today.
I love those old fashioned pieces of machinery, if a modern day appliance would break there is a good chance you would not be able to do anything about it, but this old stuff just takes some patience and basic observation skills and you can keep it going.