Any parent is going to react strongly like this to a child dying, even if they are not their only child. Especially if that child is an elf who isn't supposed to ever dye.In the films, Elrond doesn't believe in the nobility of Men, nor does he have any faith in or love for Aragorn despite Arwen having promised herself to him (Aragorn), which is why it's a significant turnaround for him (Elrond) to reforge Narsil and deliver it to Aragorn, which are actions he takes only because Arwen is dying.
If Arwen isn't Elrond's only child, the fact that her impending death is what convinces him to abandon his distrust of Men and offer Aragorn his support doesn't make sense.
Every one of their children is important to a parent, so they're not going to just shrug their shoulders and move on if one of them is dying.