I really feel for that mother! Two preschoolers, and obviously ill enough to have to see a doctor.
What time of day did this happen. I am assuming that if Noah was indeed 5, he would be in Kindergarten? I am thinking he may have been slightly younger. I know that I would try to make a Dr.s appt. when my kid was in school if at all possible.
I do have one huge problem with how the mother handled the situation. This is just two little kids. And, she very obviously expected perfect behavior, possibly behavior that should be expected of a slightly older kid, from Noah. she obviously expects Noah to be able to play 'nicely' with his younger sibling. And, when anything at all goes wrong... (Hey, three year olds sometimes grab toys and do not want to share) it seems like she is very quick to blame Noah and to expect him to handle it all by himself. Noah is to simply let the three year old do this? He is to completely defer to the younger child at all times? He is to act like a little adult?
It sounds like the mother actually exascerbated the whole situation by coming down on Noah so hard, instead of stepping in, providing a distraction, separating the two, or saying calmly, to BOTH kids, "We need to play nicely and share".
While I agree with consistant discipline. The mother did not diffuse the situation. Seems she made it worse.
Now, I know that she was ill. And I feel SO sorry for her. But, I am not so sure that this is not her usual way of handling the two kids. As I was reading this, I kind of felt bad for Noah. The way she was quick to jump on him, calling out his name very directly. It seems like this is her usual mode of operation.