If you swapped out the starter with a known good one and the problem exists it can only be one of a few things. While you looked after the postive cables to ensure good connections - did you check ground strap? There may be an extra ground strap from the motor to the chassis that may be corroded or the main ground may be corroded. Also - had one VW that drove me nuts with low current to the starter all connections were clean and tight but finally a resistance check of the cable to the starter from the battery showed that there was resistance in the cable. I cut it open to expose nothing but corrosion. New cable cured problem. Now assuming that the current flow to the starter from the battery and back to the negative post of the battery is perfect - the only other thing that can cause the problem is a missing 12 volts to the solenoid to kick it in ( using a screwdriver applies the required voltage and the spark will temporarily overcome a bad connection so it will work for a short time. Next time it won't start have someone take a voltmeter and check if there is 12 volts going to the small terminal on the soleniod when you turn the key on - if it is there then it has to be the starter or the current cables to the starter. If it is not there -
1) check to make sure the connector that attaches to the soleniod has the wire firmly crimped into it.
2) once #1 has been confirmed hold the voltmeter onto the terminal and play with the ignition switch and see if you can get the voltage to come on to start the car.
3) if that does not work - get to the connector at the back of the ignition switch pull it off the switch and check the continuity between the proper pin of the connector and the solenoid - if that is good - then suspect a bad ignition switch

That is all I can think of that can cause your problem.
Morris