Signs are pointing to your remanufactured starter. But before you replace it, check out all the circuits. You said you have a new positive lead from the battery. Are you sure you have a good ground? Try a star washer in the starter mounting bolt. Look carefully at the battery negative lead, especially where it meets the engine block, by the water pump. That is another good spot for corrosion.
If all is good, then "bench check" the starter this way: Remove the starter from the engine, and place it in a secure place, like a vise. Pull the vehicle battery, or use another one. Connect negative to starter chassis. Connect positive to big starter positive connection. Connect jumper wire to the same positive connection. Then, carefully touch the other end of the jumper wire to the smaller connection on the solenoid. That is where the start signal goes. The starter solenoid should jump forward, and the motor should spin the starter gear. If it does not, then take the starter back. If it does, then the starter works well, and look deeper into the engine, the "turn-ability" of the crankshaft, and the engine electrics.
If all is good, then "bench check" the starter this way: Remove the starter from the engine, and place it in a secure place, like a vise. Pull the vehicle battery, or use another one. Connect negative to starter chassis. Connect positive to big starter positive connection. Connect jumper wire to the same positive connection. Then, carefully touch the other end of the jumper wire to the smaller connection on the solenoid. That is where the start signal goes. The starter solenoid should jump forward, and the motor should spin the starter gear. If it does not, then take the starter back. If it does, then the starter works well, and look deeper into the engine, the "turn-ability" of the crankshaft, and the engine electrics.