Does Chinese have a gender-neutral term for "homemaker"?
持家者 = 主持家中事務的人

The homemaker = the person who does most of the housework in the family 

In my opinion, it doesn't matter it is the wife or the husband or anyone else-- who ever does most of the cleaning and cooking is the homemaker of the family-- If both husband and wife share the housework evenly, they are both 持家者

持家有道 - (good at taking care of the household) is a praise for a good wife 

The term 持家 literally means "maintain/ holding the household" which also mean all the responsibility within the household belong to that person including taking care the children.