The diode seems unrelated to the shutdown functionality. The shutdown works by shorting the control voltage to ground, forcing 0V on the output. Furthermore it also shorts the output to ground. When shutdown is in place the two diode terminals will also be shorted together through the MOSFETs so it serves no purpose there.
Indeed it is confusing that they decided to add the protection diode only when presenting shutdown functionality. So yes, you can use the shutdown functionality without it, though it's probably not the best thing to do.