Hi and welcome to the SI forum!
The basic 'No Disc' problem may be the sensor in the MD unit that tells it when a disc has been inserted after the lid is closed. It is a small black pin on the left-hand side of the MD base when the lid is open:
This pin engages with one of the holes on the underside of the minidisc itself, and the pin gets pushed down when the minidisc is inserted and the lid closed:
In older machines, this pin can become less sensitive and requires to be pushed down further.
The malfunction can be associated with particular discs though. Do you have another to try? Otherwise you can test by putting a piece of sellotape over the pin to stick it down, then put the minidisc in and try closing the lid. If it reads the disc then you can assume it is the hole depth not being enough.
I have overcome this problem by putting a tiny piece of paper into the hole of the minidisc itself, and pressing it down hard to flatten it within the hole using a small screwdriver. The effect is to raise the base of the hole and make it push that pin down a bit further when the lid is closed.
Sadly, on some units the pin gets snapped off due to rough treatment and requires replacing.
BTW, the 'white gunk' you saw on the spiral is lithium grease which is supposed to be there to lubricate the worm drive mechanism. If you cleaned off the excess then that's probably fine as there should be enough remaining in the thread to work.