I have a friend from Michigan, and he refers to his son as "the kid" all the time. I have never found it strange at all. He only has that one child.
I refer to my daughter as "the kid", however, she is 24 years old. Her step dad refers to her as "your kid" when she wants something and "our kid" when she does something nice. However, we both refer to her by her first name when we are talking to people that know her. If we are talking to people that don't know her, i refer to her as "my daughter".
As I type this, I find myself automatically avoiding her name, not because I don't want to call her by name but because I want to protect her from people that don't know her. That maternal instinct to protect our children from strangers, lurkers, and even the media could just be coming automatically for the parents during these interviews. Even though they know they need to get her name out there, the parental instinct to protect our child's identity from strangers is probably quite strong regardless of the situation.
Just saying.