Good one! This has been a long time since I work on one. LOL. The rear trany pulley was aluminum and the belt would wear the V pulley wide, then the belt would ride down to the bottom and then slip.
Back to your electrical. This has happen to me before many times, but not on this model. The thing that comes to mind is the key switch, but I don't think that's it, but look at the back where the wires plug into it. Look for burn thru at black plug.
Next would be one of the two relays. They are in a bad place to get too. Remove the battery and you will see them, one up and one down. I can't tell you witch one may be bad. I know one is for the PTO, the other I can't remember what it did.
What maybe happing is your bypassing something as long as you hold the key to start. Then let go and lose your spark....... .. ... .. .. . I'm still thinking.