Sorry for taking so long to reply. Here's the latest.
Took it into a Ford dealership to have the recalls done and an oil change, so I asked them to please plug the car in with the cord and see if the cord overheats. They said they would.
Next day (today) when the car is ready pick up, I asked if they were able to duplicate the overheating. They said no. I asked if they just plugged the charger in the wall (like the previous dealership), or if they also connected the charger to the car. I got a long pause and a "thinking look", and a response of, "I'm pretty sure we probably plugged it in". I asked for how long, and the response was "a good bit". When I took the car (which I dropped off in hybrid mode / no EV charge left), it still had no charge - so even if they did plug it in (which I doubt), it couldn't have been for any more than a couple of minutes.
I will give them credit for saying they called Ford to ask what they should do, but Ford told them "there's no warranty on the cord", and gave the dealership a list of questions to ask me. Those questions were:
"Is the breaker no more than 15/20?" (mine is 15, so this was good)
"have you tried the plug in several outlets on different circuits and still getting the same problem?" (yes)
"is the house a newer house"? (yes, built Q4 2007)
"do you have any issues when you plug another device into the outlets?" (no)
I think the tech was expecting (hoping?) we'd be non-complaint with at least one of the questions, but we were good on all of them. So he said, "well, anyway, Ford told me to tell you there's nothing we can do, so you should hire someone to come look at the problem".
Called Ford Customer Service - very nice lady who tried to help, but in the end shared that Ford doesn't have any processes, equipment, or guidance for dealerships on how to diagnose possible issues with the power cord. She mentioned that all the documentation ends up with the same instructions, which is to advise the customer to hire someone to come to the house and investigate it.
Not sure what to do next. Car gets turned back in from lease in the not too distant future, so not sure we'll keep it.