Okay, it's a collision with an existing assignment to Ctrl-V, somewhere that I can't find, and only in certain apps, depending on their built-in implementation of whatever they think Ctrl-V should be.
Example: Remapping "Copy" to Ctrl-C always worked. So I know Ctrl-C is safe. If I remap "Paste" to Ctrl-C, it works fine. Pasting with the remapped key combination of Ctrl-C works as expected.
So the problem isn't copying and pasting, it's the particular key combination. I need to figure out how to find out what the normal thing for "Ctrl-V" is in certain apps and kill it there, perhaps by finding it in the menus and remapping it for those apps.
If I turn off remapping of the Ctrl-V key combination, it seems like the page scrolls when I press it. But I can't find anything for Ctrl-V in the menus of these apps.
Still poking at it.