My ex and I had similar arguments. Neither of us was particularly great at housekeepimg, but he was retired, and I was working fulltime, and he stillexpected me to do all of the cooking (because he claimed not to know how, even though the managed to avoid starvation before we got together) and the cleaning (because he had a bad back).
My current husband is much neater than I am, but he knew this going into the relationship. I work full-time, and he is a full-time grad student who commutes from Baton Rouge to New Orleans several times a week. In addition, our whole family (me, my husband, my daughter) is involved in community theater, plus my daughter has an after-school job. So none of us has a lot of spare time to clean.
We solve it by picking a time when we are all free (like a Saturday morning) and doing the major chores (vaccuuming, dusting, cleaning the bathrooms) together. My husband is not a fantastic cook, but he is a capable one, and he has learned to use the crock pot, so that dinner can be started before leaving in the morning, and ready when my daughter gets home from school. During the week, everyone picks up after him or herself. The house isn't as neat as when my husband lived alone, but he has learned to adjust to it. As long as the floor is not crunchy, and the windows are not adhesive, he can deal.