The fact that when he was single his wash basket was over flowing ..... and his mother moved in to do it all tells you all you need to know......
What happened after - did you unconsciously take up this role when he was single, when you were living together pre kids?
He sounds like a spoilt misogynist entitled man-child. He is treating you with contempt.
He doesn’t care about you - he doesn’t see you as a team.
You need to communicate (deep breath - no blaming or shaming) and decide if you are a team and then collaborate and negotiate a plan that makes you both happy. Start strategically. Talk through your “family values” - your aspirations on time and emotions for this challenging period of your life - and how you can both equally contribute to achieve these. Maybe time for both of you to attune to your children is more important - so housework is minimised, relegated or outsourced. Same with his work hours - they will need a change to allow him to engage in the team effort and mutual support that is a family.
His response to this approach - ie what he really wants, how much he is prepared to work with you will tell you everything. If he is obstructive and resistant - then you need to call him on it - tell him that if you are not compatible then you need to go your separate ways.
What was his family / gender role blueprint growing up?
Kindness and respect is what you should be aiming at for each other.
The plate on the side is deliberately passive aggressive. Have you read the v famous article “My wife divorced me because I didn’t put the glass in the dishwasher” ....?