I believe New atheism has come under fire for mysogyny and a certain High profile New atheist at that.
As a movement, the only people that claim there are 'New Atheists' appear to be people who aren't classifying themselves as New Atheists. There is a situation where man are significantly more highly represented in the groups associated with 'New Atheism', but that's no more a facet of New Atheism than the same balance in, say, Parliament or the FTSE 100 boardrooms. The movements have arisen in a culture of institutional, relatively low-level, gender imbalance (and, similarly, racial imbalance).
There are, almost undoubtedly, individuals within the group you define as New Atheists who are misogynist to varying degrees in their outlook - are you suggesting that's because of their atheism? Or are you just making the observation that New Atheism doesn't make people perfect, because there is institutional misogyny and individual misogynists in, say, most Western Religious institutions? Or are you just firing of another ad hominem in the absence of an actual point to make about atheism?
He called his great tome 'The selfish Gene' and provides a backdrop for the social darwinianism of the Thatcher era.
I suggest you read it, if you haven't already. If you have, I suggest you try reading it again. He specifically schools against interpreting society as a vehicle for social Darwinism; that other people took the idea of 'survival of fittest' and decided it was a fine work ethic isn't down to him given that he neither coined the phrase nor advocated for its use in that context.
Atheism thus passed from being left wing and socialist to being right, libertarian, socially disinterested and Xenophobic.
Atheism has no wing, nor is it libertarian or conservative. Here you are confusing atheism and social Darwinism, as your first mistake - the idea that Thatcherism was at the same time advocating atheism is laughable.
You are also suggesting that these ideas are flowing from atheism, rather than the reality that they are associating with it or adopting it a the culture in which they are both expressed moves.
Various social and political classes might have favoured atheism at various times, but that's a facet of their politics and of culture, not of atheism.
O.