I do not believe the shaft of the shifter button has anything to do with the reverse lights. There is a combination neutral start/reverse light switch mounted on the left side of the shifter assembly. There is a small pin at the bottom of the shifter that mates with a small plastic sliding part on the switch that moves back and forth in sync with the shifter. Check to see if the sliding part of the switch is actually moving with the gear shift lever and if it is check to see if there is 12 volts going into the switch and coming out of it when in the reverse position. I do not have the diagram in front of me for an '86 700, but have one for an '83 760 and the wire that should have 12 volts when the ignition is on is blue/red and the wire to the reverse lights is black.
If you want me to look up the wire colors for your '86 just let me know. John