I don't understand how this is even remotely a take-down of the paper. There are strong effects at 9 months, several months after the court decisions shown in the graph. If any two events were going to sway large numbers of voters I would think it would be the court decisions and the media coverage that surrounded it, followed by a gay canvasser coming to your door.
So if Gelman's point is: these people would have changed their minds anyway, this just impacted the timing, my question is "when are all those people in the control going to change their minds if not after a major court case"?
If Gelman were right, and ML/DG's canvassers were picking low-hanging fruit that would have had their minds changed by anything, we would expect to see big effects early on (we do) and then the court decision would prompt changes in the control groups but not much in the treatment group--since the low-hanging fruit had already been picked by their canvassers. But we don't. In fact, the jump at the court decision looks to be biggest among the treatment group.
I understand he's trying to situate it with the prevailing public opinion literature, but I don't really get it.