Neutral Safety Switch

I never know how much the person asking the question knows. Plus when you say you disconnected the wire from the switch, I don't know what that means. I can guess, but what switch are you talking about, so here goes. The neutral safety switch (NSS) in the tranny completes a ground when the tranny is in neutral or park. There is a wire going from the inner fender mounted starter relay (left side behind the battery) down to the NSS. Another one of the wires going to the relay is from the ignition switch. When the ignition switch is turned to the start position, current flows through the ignition switch wire to the relay, through the relay, and down through the wire that goes to the NSS. If tranny is in park of neutral, a ground is completed and the current is allowed to flow through the relay to a ground. That activates an electromagnet in the starter relay that completes another circuit that allows current to flow through a wire from the relay to the starter solenoid. Then, assuming you have a good starter, the starter will turn over.
First, check to see if you have current at the relay in the wire coming from the ignition switch when the ignition switch is turned to the start position. Then check for current at the relay in the wire going down to the NSS when the ignition switch is turned to the start position. If both of these current tests are good, you likely have a bad NSS. Using a jumper wire, ground out the relay terminal that has the wire going to the NSS. That will simulate a NSS that works and allow current to flow through the relay to the starter solenoid.