I have to walk that back some. I had put the scope away, but I decided to bring it back out and have a look. Despite improved fading over the way it was before, I saw on the scope that it was only conducting on one alternation, asymmetrical. Then I plugged the 40W lamp in with it and it looked even better than with the MOC3023. It's totally textbook with the lamp plugged into it.
So, one "strand" isn't enough to do it with the TRIAC. My strand takes 20ma. My TRIAC's (BCR8N) holding and trigger current are both 100ma, which means a load of at least 12w (ballpark). Maybe 5 would do it to it, but I don't have that many to confirm.
Still, the MOC3023's aren't a bad way to go for this application.