I've read on the Battle of Allia and apparently a major cause of defeat was the fact that the Gauls were yelling out terrifying war screams that played a major role in breaking the Roman Phalanx. However this was before the Marian reforms and was at a time when the Romans were farmer-soldiers so this did not surprise me.
However I also read years after the Marian reforms, when the Roman Legions were confronted with the Gauls and other "Barbarians" there are descriptions of Roman soldiers shaking in fear at the initial phases of the battle when the Barbarians were yelling out their fierce war cries. In some cases Roman Legions were paralyzed according to various stuff on the internet that they cannot move or maintain ranks.
I am curious what makes hordes of screaming Barbarians so scary to even train soldiers like the Romans? Modern military standards would consider the thoughts of fearing an enemy force simply because it screams and yells so much as a mark of poor discipline!
Its not just Ancient Warfare. I have read of the "Rebel Yell" that the Confederate Infantry would use. Granted they were trained and organized as opposed to the Barbarian Hordes, but I'm really amused why Union soldiers would be terrified of this tactic well in fact more dangerous things were taking place like bullets were being fired. So what make warscreams so scary that they could lower morale and even make entire units collapse?
This may well be deleted, but I feel that this sort of question would be more suited to something like /r/askpsychologists (if it wasn't so inactive). I'm not sure that an historian can quantify what exactly it is that makes a battle cry so terrifying.
Fundamentally, it is very difficult to imagine the terror of combat unless you've experienced it yourself. Training will help, but faced with the imminent threat of death the human mind will not so easily rationalise the situation as you have done in your post. Given the number of times that we can see battle cries being used throughout history, it is clear that they have a powerful effect on morale on both sides.
NB: You seem to assume that the Romans themselves didn't use war cries, when in fact the barritus was commonly used in the later years of the Empire.
Sources:
T.J. Craughwell, 2008, The Vikings, Vandals, Huns, Mongols, Goths, and Tartars who Razed the Old World and Formed the New, Fair Winds Press, p. 41
M Whitby. "Morale: Late Empire". The Encyclopedia of the Roman Army, 2013
The Gauls did sack Rome early in the cities history, this may have something to do with it. As far as the later Empire, most of the Barbiarians were bigger, taller, and more unpredictable than the Roman troops. Ammianus Marcellinus wrote a lot about the Germanic tribes that clashed with the (soon to be emperor) Julian in the late 350's...