I was given this player, which I've been told is one of the few Sony consumer models really worth having, by an acquaintance. In addition to an elaborate set of karaoke functions, it has AC-3 output, and PAL/NTSC VCD playback!

The first problem I encountered was that it would not load/unload. The belt had become flabby, so that the motor would spin but not move the tray mechanism with any force. As a result, the player would shut itself back off in short order.
I was able to replace this belt with a SCQ4.6, although this is not quite as heavy as the original. Maybe SCC4.6 would be better? In any case, it loads & unloads now.

Remaining problems :
1. Does not detect CDs.
2. Detects LDs, but does not spin up.
3. CDs get stuck to the top clamp.

The laser lights, as I have visually confirmed. Also, I believe the motor works, because when I press "play" with a LD loaded, when it "gives up", there is a noise, and the disc does "twitch" slightly.
Tips on troubleshooting?

I have succeeded in obtaining a service manual (and e-mailed the file to Julien). I guess my next step is to use the "debug mode" and "service mode" to find out what errors it's reporting when it fails to spin up, or what sequence of events it's hanging up on.

Any way to check if the capacitors are in good shape? I know Duncan, when he used to post here, said that Sony players are notorious for that.

And when I did have a Sony LD player (MDP-A3), the capacitor blew for the front LCD(?) display.

Using the procedure in the service manual, I entered "debug mode" and found that the "emergency codes" all indicated a failure of the focus operation. I then proceeded to "service mode" and discovered that the command to operate the focus servo resulted in no motion of the lens. So, either there is a circuit fault in the optical block, or one on the main board where the focus drive signal is generated.
Once I build an extender board, I should be able to tell the difference, by looking at the focus servo drive voltages. And if it's a main-board fault, I at least have an idea where to look.

A-ha!
There is a place on the main board where a transistor appears to have, for want of a better term, fallen out. This is marked as Q709, which I see by the schematic to be part of the Focus Coil Drive circuit. I still need to put an ohmmeter on the ribbon cable, to check continuity, but I'm guessing that simply resoldering this transistor might fix the fault. Perhaps I should go ahead & buy a new 2SB1565, though.

UPDATE Continuity on the focus coil checks good, although the ohmmeter doesn't move enough current to deflect it. At this point, I have no reason to think that, when I replace the transistor (hopefully tomorrow), I won't have full operation immediately.

Nice to see you're making steady progress along the way to get this player working. Good luck with this next step!

Looking forward to see how this player performs!

Well, that was not the result I was hoping for.

Replaced the transistor. Hooked the player up to power. Put it into "Service Mode" to use the function that runs the focus coil up & down. Noticed a curl of white smoke rising from the optical block. Pulled out the power cord. Touched the transistor & nearly burned myself.
Two thoughts arise : firstly, the previous transistor must have gotten so hot it unsoldered itself & fell out ; secondly, although I can buy a new optical block, I first need to identify the underlying circuit fault, so I don't smoke it as well.