Before the chick hatches, it absorbs the yolk.  It can live off of that yolk for over 72 hours without eating or drinking.  That way a chick that hatches early can wait on the late hatchers before the hen takes them off of the nest for food or water.  So there is no need to take the chicks out earlier than 72 hours after hatch.  I had a broody hen wait over 80 hours after her first chick hatched to bring all of them off of the nest.  All of the chicks were fine.
Are you familiar with the term "shrink-wrap"?  That's where the membrane around the chick while it is still in the egg dries out and shrinks around the chick preventing it from hatching.  It is pretty rare, it does not happen often.  I've had it a couple of times.  Many people open the incubator in the middle of the hatch all the time and never see it, but if the chick has pipped and the humidity drops significantly it is possible.  
I don't let that possibility stop me from opening the incubator during hatch if I have a real reason.  For example, during hatch a half of an eggshell enclosed an unhatched egg so a chick would have trouble pipping and zipping.  I opened the incubator and took care of that.  No problems that time.  But since I know it is possible to shrink-wrap a chick I don't open the incubator without a real reason.  To me, lack of patience and a desire to cuddle the chick is not a reason worth taking that risk.  72 hours is.