In those countries, the wife/mother of the family doesn’t work outside the home and handles childcare and elder care inside the home.

Not taking a stand on what’s the best structure, but that’s the “somehow” you’re referring to.

People also survive without access to warm water. I think the discussion if external or family care is preferable is answered for both parties in favor of external care.

If one family member stays at home, such care would be possible, but this reality becomes much rarer today. Otherwise you indeed would almost certainly have to quit your job since care is an almost full time occupation you cannot do on the side in many cases.