I think it was perhaps Mencius Moldbug who originated the stupid idea - which I have seen repeated in hundreds of different versions - that the current, mainstream, politically correct Left are puritans.
Of course there is a grain of truth, else the idea would have gone nowhere. The grain is that New Left is a descendant of the New England Puritans who emigrated from (mostly) East Anglia, became the Boston Brahmins, founded Harvard etc.
And this class, via various mutations including the Transcendentalists and their circle of radicals Unitarians, abolitionists, feminists etc) evolved into the post Civil War US ruling class; who were the fount of post-middle-1960s New Leftism.
OK. But to call the New Left puritans is something only a non-Christian could do, for at least two very obvious reasons.
1. A puritan is very religiously Christian, and believes that this should permeate every aspect of social and personal life.
2. A puritan advocates that sex be confined to (a single, permanent) marriage. In other words, a puritan rejects the entirety of the post-sixties sexual revolution.
Since Leftists are not Christian, and since they are (in theory and in practice) sexual revolutionaries; the idea that Leftists are puritans is wrong.
The fact that so many supposedly 'Right wing' anti-political correctness commenters accuse Leftists of being puritans, simply shows that those who make the accusation are themselves not Christian and are pro-sexual revolution. In other words, to accuse Leftists of puritanism is itself evidence of Leftism!
Such an accusation could only come from someone who does not take real religion seriously, and who regards sex and sexuality as trivial.
In other words, the accusation that Leftists are puritans could only come from a Leftist.
A Leftist can concisely be defined as anyone who is not primarily religious, as a principle of social organisation Or, a Leftist is anyone who puts forward any this-worldly principle (happiness, equality, justice, freedom, prosperity, minimisation of suffering, law and order, science...) as the primary goal of social organisation.
And the easiest sign of a Leftist is: someone who is in-favour-of the sexual revolution.
Note: At least in the USA; the use of 'puritan' as a term of insult seems to have been popularised among the intellectual elite from the socialist, pro-communist left of the early 20th century; such as Van Wyck Brooks's influential 'Wine of the Puritans' of 1908.