I agree on poorly re-manufactured starters being out there. I have also gone through a bunch in the past with my ex wife's LTDII.
I would like to add that ford flywheels will over time burr up on the teeth, and cause a motor lock up! The flywheel can be pre grinded at a angle for easier mesh-engagement. Your problem sounds like a rare factory machining problem. You could use a dial indicator on the flywheel run out---rotation. You should remove the trans. and install only a bell housing with the starter mounted on the engine. Have a sloted viewing window on the bell housing for inspection while cranking to check starter engagement --- aglinement. These are processes of elimination, I wish you luck and wisdom. This is why I still drive old Fords If needed I will elaborate further. Happy fathers Day
Any idea if the transmission or the engine was ever out of the car? I had my head up my #$$ one time when I was installing my engine. I bolted the flex plate to the converter and started her up. Nasty grinding noise and the starter was locked against the ring gear. Turned out that I did not align the converter drain plug with the hole in the flex plate. This warped the flex plate. Remove the inspection plate and rotate the engine. Look for a warped or cracked flex plate. Take a look at the ring gear too. Micah