After reading this over with my GF, I think I see what you are getting wrong.
Post 1 from sdifox.
He questions the usage of puritanical.
Further context given in post 2 from sdifox.
What I think you and sdifox are having trouble with, is the association of puritanical with the puritans. So, a group may have traits that may not be connected to other traits beyond being part of that group. The adjective puritanical has this definition "practicing or affecting strict religious or moral behavior." That means something could be puritanical if it is religious or moral. I believe you are getting stuck on the fact that the Puritans were a religious group and thus any connection must also be associated to religion. I believe this is sdifox's belief as well as he references things that are part of the Religious Right such as pro-life. He also says that Puritans left because of liberals, so to use puritanical to describe the left would be incorrect. That is, however, not how words work.
For example, think of Nazi. Someone can be a grammar Nazi. Grammar is not a political issue inherently, but when the term is used, it does not imply anything political. That is because Nazi in this sense is used to describe someone that is very strict. This is also how puritanical is being used. To call someone puritanical-x is to say that they are either very concerned with religious OR moral behavior.
So when Glenn said puritanical nanny-statism, he is saying that there is a group (in his opinion the left) that is practicing strict moral nanny-statism. His usage is correct, and just because Puritans would be considered on the Right today, does not change that the adjective puritanical can be disassociated with religious. We know this by its definition.