Are the Nano V3's especially fragile, or am I just unlucky?
I would say it looks like the Nano is a more fragile design. The standard Arduino board uses a thermofuse to limit current from the USB port to 500ma and resets automaticaly when the short is removed. On the Nano there doesn't appear to be a thermofuse installed. The burned component looks to me to be the D1 isolation diode. It's used to 'switch' the +5vdc source from either the USB power source or the on-board +5vdc regulator and allows whatever source has the higher voltage to supply the boards current. The parts list shows it as a 500ma diode, so most likely a careless wire could cause a short circuit and is burning the diode open.
Just a guess as I don't have the Eagle program installed to actually see if that component is D1 or not.