How do I figure out what crankset replacement will fit my derailleur?
According to the Bikepro.com bottom bracket compatibility page, the M560 crank requires a 113mm x 68mm bottom bracket (113 mm is the spindle length), which is in agreement with your service manual (113 mm for a 47.5 chainline, or 118 for a 50 chainline).

The Shimano FC-TY301 crankset requires a BB-UN100 / BB-UN26 bottom bracket. The BB-UN100 only comes with a spindle length of 123 mm. 

What this means is that you're using a bottom bracket with a spindle length of 113 mm with a crankset that was designed to be used with a bottom bracket that has a spindle of 123 mm. Since the entire crankset it basically shifted a whole centimeter towards the frame, this could be causing the problems that you are describing.

Usually you can get away with using a different bottom bracket with a slightly shorter or longer spindle, but it appears that in your case you require the correct bottom bracket to make everything work properly.