I opened the project in Platinum and the same thing happened.
I tried the same procedure on a different track (back in CbB) and it worked the way it's supposed to, without the fade. What a relief.
As an experiment, I created a new track to paste the section to. It showed up without the fade. I copied that clip and pasted it to the original track and it behaved properly, without the fade.
Most peculiar. At least I have a workaround if it happens again.