If she opts out of the paid workforce, they both are not breadwinners, by definition.
Currently, they could make it on his salary alone, but not on hers alone. That also pretty much makes him the breadwinner. But it's not fair to lump all of the financial responsibility on him, without mutual agreement. Nor is it fair for her to do most of the childrearing and housework.
But I do think people invent busywork in the home to justify being an sahp. There is no need to have excess possessions, to create "must do" things like birthday parties, seasonal decor, excessive kids activities, etc., to fill the time. Focus on the children's education and simple family time with a minimum of possessions.
They should review all of the domestic tasks together, decide which ones can be jettisoned or how the house can be decluttered/streamlined to make maintenance easier, decide on whether to outsource anything (cleaning, gardening, hire someone to do school runs, etc.) and then divide up what is left.
If he won't help, the obvious is to drop the housework to bare minimum for herself and the kids, NOT to drop out of the paid workforce. That is the road to ruin.