This person’s whole point rests on people taking every word of the scripture absolutely literally - and if we did which we pretty much never have, there would be no need for the church or its teachings. Show me how all of Christendom doesn’t eat seafood or wear polyster cotton mix clothing for fear of being struck down and then there is a case to even suggest half of what that individual is saying.
There is nothing misogynistic about Christianity, misogyny is hatred. A Christian has issues with it if they can be persuaded by the scornful to take hatred from the faith of Jesus Christ, to be honest.
As for the more obviously observed bits being “sexist” well anything that differentiates between the sexes could be called “sexist”, and the sexes are in fact different, so we could sit here all day coming up with examples of why life itself is sexist, why families are sexist, why actual sex is sexist, to demonstrate why submitting to your husband in marriage could be seen as horrendous and sexist.
Notice though that he missed out the equal and opposite part which says a husband must give up his life for his wife to make that point. Could be seen as pretty sexist too, unless you see it as what it is - a call to be devoted to your spouse.
Perhaps it would benefit this person to have more than a glancing idea of what he is talking about - especially with regard to the history surrounding and immediately preceding the spread of Christianity proper. The things Christianity brought the the ancient world - in terms of the treatment of women and respect for marriage most particularly are to be regarded as far from evil and oppressive.
Honestly, that is the sort of person that wants the rest of us to answer to them for our faiths, so they can arbitrate over us all and The LORD with their great wisdom and intellect. Somebody needs to be praying for that man right now.