I am a father of three. Were I in those horrible shoes, I'm quite certain I would feel that my wife was the one "being told". Maybe that's just some misplaced "macho" attitude on my part, but it's always been true, in my experience.
So, I am very sensitive to your point, and I think it's an important one, but i don't think it's the primary motivator for the use of "mother" in the article.
I suspect the reasons are (at least) twofold. The primary demographic served by this ER doc has a very high rate of single mothers. I'm not making commentary on that beyond the simple fact that it is the reality. My guess is most of the time she's talking to the mother because the mother _is_ the only parent present.
When the father is there though, I studied he feels as I do... In that awful moment, I would want the doctor's attention on my wife. I would want as much support as possible for her in that time. I'll make do on my own for now... Like I said, that's likely misguided chivalry or a 'macho' attitude on my part, but it's my honest take on the situation, and I suspect I'm in the majority.
I normally wouldn't bother with a follow comment about a typo, but I think it seriously impacts the readability (like I said... feeding twins in the wee hours of the morning...)
It's abstractly difficult to inform a person that their close family member has died, potentially from violent causes, possibly very suddenly.
But come down this hallway, now, to tell this woman that her child is dead; this is what it will be like. You'll need to change your clothes first; you'll need to practice first; and this is what it will do to her, and what it will do to you.
I (obviously) found this very powerful, and while there's some compromise involved in adding in these carefully-selected specifics, it's worth it.