Respectfully disagree with your statement.
Sometimes in serious issues it is way better to say "No problem, we will allow you to change your ticket, change fee free and free of fare difference", but Delta will ask for treating Doctor name and address of hospital. Will they cross check? Sometimes yes, sometimes not.

The DL CS desk is always helpful and waived change fees and got pax or relatives early/late.

I too hope the OP's child is ok and strongly recommend that she calls the medallion desk. And truthfully states the underlying reason for the change request.