In theory, left-handed warriors could have had an advantage in ancient warfare, but not for the reasons you suggest. Ancient battles were not won by superior swordsmanship, and a single man's prowess did not have any impact on the outcome of wars. These battles were an activity of large numbers of men, typically fighting in formation. With that in mind, the important difference between a left-handed warrior and the more common right-handed one would have been that they held their shield in the other hand.

In marching columns and battle formations it was taken for granted that everyone was right-handed. Each man would provide an identical form of cover for himself and the men around him, and arrayed side by side, they would form an unbroken front. But the ancient Greeks already realised that this transferred the weakness of every man to the whole formation: as he marched, one of his sides was well protected, but the other side was not. With every man carrying his shield on his left arm, every infantry formation had what the Greeks called a "naked side". An army on the march could not protect itself effectively against attacks from the right, since its shields were all on the other side. In order to block even a simple volley of rocks or javelins from hidden assailants, they would need to stop, turn around to face their attackers, and keep still with that facing until the enemy were driven off, lest they expose their naked side again. In battle, too, the right side of the formation was especially vulnerable to flanking attacks, since individual warriors had no way to protect themselves on that side without disrupting their formation.

This vulnerability was also exploited in the design of fortresses and city walls. The most obvious points of attack in these fortifications were the gates, and so many city gates were not placed in line with the wall, but perpendicular to it - and facing left (when seen from within the city). This forced the enemy to turn left to approach them, exposing their right side to the defenders on the walls. A common elaboration of this defence was to extend the wall past the gate to create a corridor, which the enemy would have to funnel themselves into to reach the gate, keeping their unshielded right facing the city the whole time. Many fortified places would be designed to ensure that anyone who tried to approach the gate would spend as much time as possible with his shield uselessly facing outward, while his "naked side" was open to attack.

In theory, then, the left-handed warrior might have a real advantage, both on the march, in pitched battle, and during siege assaults. There would have been various occasions on which a hypothetical unit of men carrying shields on their right arm might have helped to protect the whole army.

But no such units were ever created. To my knowledge, it was never even considered. I'm not an expert outside of the Greek world, but speaking for my own specialism, there is simply no discussion of left-handedness as a factor in Greek warfare at all. Not a single text ever mentions it. The right-handedness of all warriors is absolutely taken for granted. I've answered a question similar to yours on this subreddit years ago, and I've been able to link to it many times since, because it seems a lot more redditors have considered this possibility than people in the ancient world.

As I mention in that post, there is one instance in the Biblical Book of Judges in which the Benjamites raise a unit of 700 left-handed swordsmen. However, the only notable feature of these men according to the account of Judges is that they were very skilled slingers. They never recur in the ensuing narrative of successive battles against the Israelites. There is no indication of what these men did in these battles, or even why their unit was specially created. We can speculate, of course, that this is just such a left-handed flank guard, meant to protect the right wing of the army by holding their shields on their other arm. But that is no more than a theory, and the experiment is never repeated.

We can think of some obvious reasons why people wouldn't be interested in pursuing the idea. The main period in which an army of right-handed men would be vulnerable was during the march. But how could a single unit of southpaws correct this? Marching columns could only be as wide as the road they marched on; you couldn't simply add a whole extra unit along its width. In any case, you wouldn't have enough left-handed warriors to form a unit as long as the whole army column, so only a short section of it would be covered. The alternative would be to string the unit out along the entire column in single file - which might provide the protection of its shields, but would make it impossible for the men to function as a unit. It would take them perhaps literally hours to draw themselves together into a battle formation of an emergency called for it. In other words, either you spread the men out to the point where their presence becomes a logistical liability - how do you ensure that every stretch of your regular battle line has its assigned detachment of exactly as many left-handed men as it needs? - or you gather them together in a way that renders them unable to do most of their assigned tasks.

It might have been conceivable to create a unit of left-handed men for the purpose of holding the right flank in battle; perhaps if you bestowed the right kinds of rewards on them you might even be able to get them to lead the charge against every city gate that forced a left-turn approach. But nothing like that was ever attempted in reality. Almost certainly the process of selection for such a unit would make it too cumbersome. There is also the problem I noted in the linked post: to compensate for their vulnerability on the right, hoplite formations tended to edge to the right as they advanced. Imagine putting a unit on the end of the line that edged to the left instead - its vector crossing that of the rest of the army. Having such a unit would cause nothing but confusion as the battle line sought to maintain a front.

In any case, this is all idle speculation; the most important reason why these units never emerged was probably that there was considerable social stigma against left-handedness in many ancient societies. The prejudiced association of the left with bad, crooked, and evil things (for instance, the English word "sinister" comes from the Latin for left) may have nipped any suggestion of such a unit in the bud. Instead, the warriors of the ancient Mediterranean world apparently universally fought with their right hand, and carried their shields on their left.