"Why would I be so irritated by the way a child sounds when he breathes, especially when he doesn't have any control over his nose being stuffed up like that?"
The first is simply a conditioned responce. You have come to assosiate the sound with the child. Just as Pavlov's dogs assosiated a bell with eating, and started salavating as soon as they heard it.
The second is that to a certain degree, he probably could do something. Has he ever asked for a tissue? And you know it isn't shyness holding him back from requesting a tissue.
The third is the least fair, it's human instinct. Even before the dawn of humans those who avoided others who showed outward signs of illness were less likely to catch illnesses and therefore more likely to survive. Instict tells us to avoid the sick, it is only our modern human intellect that allows us to over come these instincts and take care of the ill (our own children are an exception to the instinctual avoidance, since they are the carriers of our genes, and therefore an extention of our self preservation instinct.)