Original wiper motor was locked up tight so I ordered a replacement motor (CH28545) from OPGI. Got it installed with minimal fanfare but it won't start on its own. I have to give it a nudge to get it to rotate. Low speed it will only rotate once and stop. High speed will continually rotate until I turn off the switch. Will not restart from the switch under any conditions. Any ideas?

Did you check all the linkages? Something is sticking and the motor can't initially overcome it. Your car is getting close to 60 years old and it's things like this that can remain forgotten about. If you disconnect the linkage from the motor's shaft and it works fine, you know there's a deeper problem. It only takes a broken washer or a worn bush to cause havoc.

The motor is disconnected from the linkage in the attached video. Seems to me that it isn't turning fast enough to start. Is that even possible?

I would question if the switch is at fault , pull the motor and bench test it without the switch in the circuit

But make sure you have a good 12 volt supply too !