I went ahead and tasted one of the Black Nightshade berries about an hour ago. I didn't ingest it; I chewed it up and spat it out. I rinsed my mouth out with water afterwards, too.
It wasn't good, but it wasn't bad, either. Very weak flavor. Almost flavorless, really. Just a slight tartness like one might find in a raspberry, but much less strong. No sweetness at all. More similar to a tomato, I guess, than to other berries, which makes sense since tomatoes are also a nightshade plant.
The berry's skin is very thin. It crushed easily when I chewed it, much easier than a cherry or grape.
Anyway, my curiosity is satisfied, so that's the end of that. I went ahead and pulled up the plant. There's a little sprig of what I think is wild blackberry about a foot away from where the nightshade was, so I'm going to see if I can train that plant to take the nightshade's place. And hopefully this time what I think is wild blackberry will actually turn out to be just that.