Why change the word though? She says hate, she means hate. As a journalist, I accept her ability to write what she means, to understand what she is writing. After all, it's a powerful word.
In paragraph 7, she uses it again - 'men as a group certainly do (hate women)'. There is no confusion as to what she is trying to convey. This belief is fundamental to her article. I disagree with her.
On the point of calling out sexist language or behaviour - I could not agree more. Her reasoning about why it happens though - that is where I think she is wrong.
Lastly, your point about paras 2 and 3 - I am not asking anyone to modify their language, to pander to my feelings or to spare my injured male ego. I get why feminists pursue this line, but it doesn't anger me. If anything it makes me feel a bit sad.