In the first post it was stated that there was power to the starter and to the solenoid. Also when the key was turned to start the amp gauge showed a drop. This would eliminate the neutral switch and confirm a complete circuit. It would also eliminate a bad ground. This is a common problem on cars with headers, exhaust running close to the starter, or weak solenoids. Also known as "hot soak" condition. The starter just won't work until it cools off. A weak solenoid has the same symptons. Other causes can be the plunger binding or dragging inside the solenoid, too stiff spring on the plunger, worn out pivot pin or linkage, starter drive sticking on armature, or an overheated winding. In most cases tapping on the starter while holding the key in start position will get it to work. I go thru this scenario at least once a week at work.
Testing a starter on a bench is very inaccurate because it does not duplicate the same conditions. Jumping the solenoid with jumper cables or a screwdriver has much less resistance than a 12 ga wire, 10 feet long running thru 2 switches, one bulkhead connector, and a fusible link. You cannot put a load on the starter as spinning an engine does.
I am assuming that Ed's original testing was accurate and that his wiring is in good shape. I don't know if he was reading actual voltage or ghost voltage. Without knowing what the amp draw readings were on the wires I cannot be 100% positive. I am going by 28 years of experience and the most common causes.
Mike