That's a pretty tendentious view of both Popehat's expressed opinion and the NYTime's article.
Popehat is pretty clearly against what he defines as "cancel culture," and the NYTimes article is primarily about journalists and elite opinion writers not liking backtalk from their readers.
Seems he is against it until conservatives are victims of it. Then suddenly it’s a good thing. You have a right to not patronize someone who you disagree with. Just like I think there will be a backlash against those who call for the pitchforks - they’ll find people peacefully protesting them outside of the their workplaces, places they frequent, outside their homes. Just another ACLU shill in this article.
One of his cited examples of actual "cancel culture" is the shouting down of Ilya Shapiro, an act he characterizes as "fascist and contemptible," and two others are either cancelings within social justice spaces or involve circular firing squads on the left. To me, this is maybe an indication that he is operating on principle more than politics.
Popehat is pretty clearly against what he defines as "cancel culture," and the NYTimes article is primarily about journalists and elite opinion writers not liking backtalk from their readers.