So he wont even cook when it is his turn to cook. It isnt about not knowing then is it? It IS laziness and misogyny. He is insisting the mental load and home running is your job and even if you give him part of the role he will do it so badly you wont ask again.
So you sit down with him with no distractions. You explain how his laziness and sexist attitude around the house is making him unattractive to you and you have considered leaving the relationship. If he doesn't change his attitude towards you and his children and his responsibilities to the family, you cannot see yourself having any respect left for him.
Ask why he thinks everything about the home and family life should be on you. Expect every answer to be sexism. His mum did it. He works hard. He doesn’t see it. All nonsense. Of course he sees it. He just ignores it as he considers himself above it.
Together write up all the daily, weekly, monthly, yearly jobs that need doing. Ensure he contributes and you don't just write the list, as he will actually need to think about them. Then you’ll see he does know what they are. Not just chores. Everything that makes up family life.
Make sure that list contains all elements of parenting: feeding the children a balanced meal, teaching the children key life skills, washing the children, reading to them or with them every night, buying their clothes, sorting their school bags, checking their homework, finding interesting days out, finding holidays, enabling their hobbies….
Ensure you have buying gifts for all the family, birthdays and Christmas. Sorting birthday parties. Sorting Christmas.
Then ask what he consistently does now from the list to contribute to family life.
Then, if at this point you still find him attractive and want to continue the relationship, and you're not angry and done with him because you've seen, written down, how poor a husband and father he actually is, you can give him a highlighter and ask him to highlight what he will now be responsible for.