Welcome, Steve, to the Corvair Forum!
You did not say whether or not you have much mechanical experience, electrical troubleshooting experience, etc. But it sounds from your activity and comments that you do have some experience in this area.
Since you just acquired your Corvair, you may not have much in the way of shop manuals and technical references. Many of these publications can be downloaded for free. The factory 1961 Shop Manual and the 1964 Supplements should help you with troubleshooting and fault isolation on your Corvair. Since you have the starter out you can bench check the starter for proper operation. Wiring diagrams and troubleshooting procedures in these manuals (attached below) should help.
The aging wiring harnesses and multi-pin connectors are often a problem, as the metal pins inside the connectors can become corroded, loose or bent. You should separate the two halves and examine the suspect connector to make sure that the wiring connection inside is making good contact.
The two-wire connector you mentioned extends the starter solenoid connectors to the engine compartment. The 12 GA PPL wire is normally tied to the harness that routes to the ignition switch. When 12V DC is applied to the 12 PPL wire coming from the starter, the solenoid should engage the drive gear with the large ring gear (clutch or torque converter) and at the same time the solenoid applies power through the thick battery positive cable to the starter motor. HOWEVER, it is important that the other half of the circuit (GROUND) is also in good condition. The ground lead from the battery must be securely connected to the engine — normally at the generator support bracket and at a stub protruding from the vehicle chassis. There are a few ways to do this, which are illustrated below. The second picture shows a flat braided ground cable that is "daisy-chained" from the battery ground terminal to the chassis ground and then on to the engine. Sometimes this cable is replaced, and the person installing a new cable only runs the cable to the chassis ground — omitting the ground connection to the engine. Since the engine is isolated from the chassis by rubber engine mounts, such a cable hookup may not provide an adequate ground return path from the starter motor to the battery. So be sure to check both the positive connection to the starter and the ground return from the engine.
If the starter appears to function OK with a bench check, and the battery positive and negative connections are good to a fully charged battery, You can test the starter function using a remote starter switch.
This type of switch allows you to temporarily bypass the ignition switch, by connecting one clip to the positive battery terminal and the other clip to the 12 Gauge PPL wire that is routed to the "S" terminal on the starter solenoid. Applying voltage to the "S" terminal in this way should engage the solenoid and crank the engine. Note that the other wire that comes from the solenoid is an ignition coil resistor wire bypass. When the solenoid is engaged it temporarily feeds 12V DC via the 20 B/Y wire to the ignition coil, which provides a "hotter" spark while the engine is being cranked (to promote easier starting). When the solenoid disengages, this 12V DC feed to the coil goes away, leaving only about 6-7V DC to the coil from the ignition switch (through a special resistor wire). The lower operating voltage through the coil prolongs the life of the ignition points in the distributor. For this reason, you would normally want the 20 B/Y wire from the starter solenoid to the ignition coil to remain connected if you are cranking the engine using a remote starter switch. This may be the reason for the short black wire you mentioned that was spliced to the 12 PPL wire that triggers the solenoid. The short black wire would allow you to temporarily connect a remote starter switch without unplugging both wires that pass through the two-pin connector.
In any case, you can test the function of the starter by bypassing the ignition switch using a remote starter switch. If the starter functions properly using a remote starter switch, but fails to function properly when turning the ignition switch key, your trouble is likely in the wiring harness between the engine compartment and the instrument panel, or a faulty ignition switch itself.
The following Corvair Forum link can provide you with a list of useful websites that should be helpful as a Corvair owner...Common and Useful Corvair WebsitesCorvair Forum http://www.corvairforum.com/forum/viewtopic.php?f=225&t=6007Left-click on each image to enlarge for better viewing...
Finally, I would like to invite you to expand on your initial post and tell us more about yourself, your Corvair, and your goals for your Corvair. If you can describe your personal assessment of your mechanical skills and abilities, that would help a lot. Members of the Corvair Forum love to be helpful in assisting other Corvair owners with technical support and advice, but it helps a lot if we have some understanding of your technical background and mechanical abilities, Corvair-related knowledge, etc. Helping us to know more about you and your Corvair will help us to write comments to you that are tailored to your needs and experience. Pictures are great too, because pictures of your Corvair will help us to visualize where you are with your Corvair and its condition at the present time. Knowing your location is also useful, because knowing where you live can sometimes suggest possibilities to resolve some issues or problems.
I hope that the above material helps with your starter troubleshooting. Please let me know what you discover or if you have any questions.