How does the head movement mechanism work in this drive? Normally a worm gear is used which is more or less impossible to damage.

It might be looking for track 0, which is located by an optical sensor, usually near the back of the floppy drive. Maybe this sensor is broken/dirty or not connected properly?

move up your drive's yellow thing a little.(do not bent) use a cotton swab and wash inside.

in fact, I revived my 5.25 fdd with this way, but $1 wet wipes not a cotton swab 😀 .

I cleaned it like that using some PC cleaning liquid (very good stuff) yet it still does that.
I think that the teeth on the moving part are damaged at the begining.

It operates by the electric motor having grooves that fit where the teeth are and by rotating it moves the slider. My theory is that the teeth are dull at the start and thats why it can not make a grip on them.
I will try to clean the sensor too.