Well, I'll tell ya. I could have sworn I'd seen this behavior when I'd gone through the SIK tutorials, and so I hooked up experiment #6, which uses a light sensor to dim an LED using PWM.
And the damn thing worked exactly the same on every PWM pin on the Uno, and on six or seven PWM pins I tried on the Mega.
So obviously, my memory or understanding is faulty. I'm quite enthused now about trying an external power source as tinman13kup forcefully suggested 
There clearly appears to be some effect due to the fact it's a motor hooked up.
I had contemplated the different frequencies of the different PWM pins being an issue, but couldn't find a correlation. But from what I could find, pins 6 and 7 are on the same timer. Which, now that I think about it, may not mean they run at the same frequency? Hmmmm. It's surprisingly hard to find information on the default frequencies for the mega PWM pins.
I'll let you know what I find!