Those old synchronous motors sometimes struggle with sync-ing due to worn out bearings or a wrong running capacitor value. I found out that the motor almost locks best with 4uF, but it slipped a pole around 4 times per second.
A video suggested to raise the voltage a bit with a Variac to help the old lady sync, and it helps!

I cleaned up the cable connections and have done the final assembly.

So it's working now. Next step is the PC interface that generates a suitable video signal, so I'll be able to actually put something onto the screen! The input signal in this video is just the mains AC taken from a tiny wall brick transformer.
I will use the same transformer to synchronize the transmitter.
The disk is perfectly locking to the mains frequency now.
The motor has four poles, so there is a 25% chance it will lock into the correct phase. However, if it doesn't, you can tap a button to let the motor slip and lock into the next phase. It's been done like that in the 1920s as well. You should have a stable picture after at most 3 short taps.