In our household, I do the laundry, dishes, and basic cleaning. She does the cooking as my cooking is "not great".
There always has to be a balance of adult household work amongst everyone.
If you both are working it becomes even more paramount that he steps up, or else well your solution is the right one.
For a reference, I am the sole breadwinner and pay 99.9% of all the bills and support my wifes business.
So if I followed traditional tropes of masculinity and relationships, I should be the king but I don't believe in that ... and neither should you.