He could of just explained the hadith and ayaah without injecting his personal opinion.
Shaykh Bakr Abu Zayd (may Allaah preserve him) said: This is one of the beauties of sharee’ah, because calling a person by his father’s name is more appropriate for knowing who is who and telling people apart. The father is the protector and maintainer of the child and his mother both inside and outside the home. This is why the father mixes with people in the marketplaces and takes risks by travelling to earn a halaal living and strive for their sakes. So the child is given the name of the father, not of the mother who is hidden away and who is one of those whom Allaah commanded (interpretation of the meaning)
Not arguing with the actual reasoning behind the hadith though, just the presentation by this sheikh. And Feminism isn't a one-size-fits-all ideology btw.