I definately second the Singer light machine oil for ball bearings where the lube has dried out .
Its unlikely to cause any compatibillity issues with the grease left inside the bearing either,
Small cassette type capstans dont tend to have a ball bearing but usually a sintered bronze bushing (possibly self lubricating) ,
Im not sure what the appropriate way to deal with them is , using a spray oil product isnt a good idea in a tape deck , you could try dropping a tiny amount of light oil right where its needed with a pin .
Again Im not sure of the exact detail of the tape mech used in that machine , you could have a drive belt issue going on , in a few other rare cases I found tape deck motors themselves were defective , there was no outward signs of any damage and they might appear to be working fine for a while but then for no apparent reason it would go off on a wander speedwise . One other thing probably worth checking is the preset pot that sets the speed .
You might also have a layer of oxide built up on the pinch roller which is causing the tape to slip sometimes , cotton buds dipped in iso should remove all that easily ,although it might require several goes at it to get it properly clean if theres a heavy build up . Examine the capstan and pinch roller for hairs or other debris caught up and wound around it . Also bear in mind some parts have plastic bearing surfaces , like the pinch wheel , you dont want to put anything petroleum based there , silicone based grease is the usual thing used as far as I remember .
Working on the mechanisms of these logic controlled transports isnt all that easy , you may have luck finding a youtube tutorial to help you .
