It's named a "mother's room" because it's for feeding/pumping, and so it's an attempt to offer privacy to people lactating. He's objecting to the name's failure to properly convey that, suggesting that (like putting changing tables solely in women's rooms) it reinforces the (harmful to both genders) assumption that childcare is a female task.
Which is to say - a "baby care room" is also a good idea, but it would be a room with a different use. The proposed name change was about making the name conform to the current intent of the room.
It's named a "mother's room" because it's for feeding/pumping, and so it's an attempt to offer privacy to people lactating. He's objecting to the name's failure to properly convey that, suggesting that (like putting changing tables solely in women's rooms) it reinforces the (harmful to both genders) assumption that childcare is a female task.
Which is to say - a "baby care room" is also a good idea, but it would be a room with a different use. The proposed name change was about making the name conform to the current intent of the room.