I mean... a guy should share the house work, it's not submissive, it's nice. That said, the usual approach to headship is female subservience. Let's be real, that's what wives are told to be: subservient. They are encouraged to be submissive and in subjection to men. That's from The Bible Teach book.
I don't want to get on a high horse about that but... The Governing Body does not want submissive men. They want men who will completely dominate their wives.