Sony CDP-C201 skipping

A temporary fix for a stretched rubber belt is immersing it in boiling hot water for 30 seconds or so in the hope that the rubber will shrink.

That may allow you to confirm that the belt is the source of the skipping problem before you go to the length of seeking out a replacement.
 
It reads the disc just fine, instantly in fact

That indicates to me that the laser power is fine.

You could destroy the laser diode by attempting to adjust the power.

I actually just tipped the player a little and it plays perfectly on an angle, but it skips when I put it back down.

I would focus my attention on why the player works perfectly when tilted.

That suggests to me that there is mechanical problem.

I have only ever solved mechanical problems with CD players. Beyond that I can only refer you to the service manual.