I've been reading through the manual to troubleshoot why the starter switch might be causing the relay to go nuts. The ZX600RC1 manual referred to a Base Manual for going through the starter motor removal in detail (which thankfully was updated on this forum 12 days ago).
After reading through the base manual, I found some possible scenarios which may be causing the symptoms it's having. A winding on the starter motor might be open (having infinite resistance) which would make it very difficult for the current to make the starter motor turn. Also, if there's a short between the commutator and the shaft, that would indicate that the armature has a short and would cause weird things to happen to the starter circuit relay. (The starter circuit relay is different than the starter relay, because it's behind the starter lockout switch. The starter lockout switch ensures that the vehicle is in neutral and the clutch is engaged - the clutch handle is out - in order to prevent the bike from being started in gear. The starter relay is what directly controls current to the starter motor).
This picture will help explain: