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From your description, there was no aparent reason to replace the starter solenoid... that acts like a switch & all it is doing is the same as you did with shorting across the large terminals with the screwdrivers...
The fact that the starter will operate when jumped directly from the battery to the starter indicates that Both of those are okay.
The Only thing left is the Positive battery cable, you have corrosion on one end or the other, you have 12 volts to the solenoid Until you short across it, then the resistance at the corrosion will not allow enough current. The ground cable 'May" also need cleandd up.... :smile:KennyV