Starter grinds when cold

That is probably a good idea. There isn't really anything you can do to a sealed starter drive unit, but you can clean and lubricate the shaft it rides on.

I believe the most we ever used to put on the shaft was a little bit of Lubriplate, but there are probably better answers now.

I would not be so concerned about trading your starter in and perhaps getting one that was rebuilt instead of the new one you were lucky enough to get the first time. The fact is that the original starter on the vehicle evidently did not grind and this one does. So perhaps it doesn't line up properly. Some starters need shims to get the right clearance. I'd just try another starter. The last thing you want is to damage a flywheel/flexplate.

I should have mentioned that the starter never grinds during cranking but just when the key is released and the engine and weather are cool, so I'm thinking that the problem is with the overrun/Bendix clutch. I don't mind taking it apart, but I'm having problems finding a manual that mentions how to lubricate it.

I bought the starter from Pep Boys only because all the real parts stores were closed late Sunday, and Pep Boys was the only discount chain that stocked a brand-new rather than rebuilt starter. I thought that not even Pep Boys would sell a junk brand-new one. :(

Sometimes there is more than meets the eye about these things.

I had one Ford that ate starters regularly, and finally found that the flywheel was warped. The car had been in an accident which might have caused the warping.

A chewed up ring gear can also deal misery to the starter drive.

Occasionally, you will get a starter drive which is no good fresh out of the box. (starter drive ~ bendix).

When you said the old one burned out, made me wonder if something else is at work on this engine.

You can go through a barrel of Chinese pliers. Most will be so tight you can hardly squeeze them. A few will be so sloppy that the nose will twist when you grab something. If the barrel has not previously been picked, there will be that one that feels just perfect. You did not get that perfect starter.