I have just finished Hayburners and had the same experience with the #3 horse- turned out to be a weak pop bumper spring/ misaligned switch stack. It was so frustratating- the game would be normal with the playfield propped up but when I layed the field down- the weak spring allowed the to #3 to make and the horse just ran forward- dumb luck that I found it- but maybe your situation is similar-

Was the switch staying closed?

Sorry- I was just looking for clarification as to the issue with the switch at the score motor (not opening/ damaged switch/ broken contact-wire)

Having had a similar issue with an animated feature of a game- if the issue comes up again I'd have another place to look.