As a point of personal interest, I would love to see your source for this.
The earliest dated cave paintings of human against human violence depict rows of archers firing at each other, apparently led by some shaman or such. They're also around the point where bows and arrows appeared.
A text mentioning such depictions, dated 30,000 to 20,000 years ago, including some of people apparently dying from arrow wounds is for example here:
http://history.eserver.org/neolithic-war.txt
The earliest mass grave dated is
Cemetery 117, and almost half the skeletons seem to have been buried with arrow points in them. Four skeletons even have the tips embedded in bones.
I have an interest in swords and sword development.
Well, swords are cool, so I can't fault you there.
But swords are the Johnny Come Late of the warfare scene, sadly. Even the first daggers only appeared in the Neolithic, anyhwere between 12,000 and 8,000 years ago.
As practical implements of war, something that could properly be called a sword didn't even exist until the Bronze Age, in the form of those early bronze rapiers. (Named so because of the way the blade was fixed, making them ok for a strictly stabbing thrust, but which would break if you tried to swing them. So basically they were mostly thrusting weapons, like the later rapier, geddit?

)
At any rate, clubs, axes and spears were already dominating the scene long before swords.
I know swords (or their ancestors) arrived on the scene fairly early, in the form of pointed sticks. There are also examples of swords which are little more than bits of sharp stone embedded in a wooden shaft.
Reference: Richard Burton's First Book of the Sword, 1883; Figure 31, pp 26-27. pdf available at the link (it works in Canada, at any rate)
More like "their ancestors" _IMHO_.
Most people would call those very early flint and wood implement "axes", "spears" or "javelins", rather than a proper sword.
And the shaft with rows of stone shards, as made famous by the Aztecs, are IMHO closer to what would later be called a
morning star in both concept and usage, than the proper blade-as-THE-weapon design that was a real sword. Those shards were more like sharp points to be driven into an opponent by impact, like in a morning star, than something really forming a fast and deadly cutting edge. Plus, at least some seem to have had more than two rows of those.
It may seem like nitpicking, but basically that's what trying to classify stuff is. If we're generous enough to call anything designed with murderous intent a sword, well, we're IMHO not going to be arriving at something particularly insightful.
But I have not looked into the earliest history of weapons in general, and thus do not know what came first. Slings, clubs, pointy sticks, bows and arrows, etc.
Technically, the pointy stick came first, even long before Homo Sapiens or even Ergaster. Followed by stone-tipped spears and axes and much later some sort of darts or javelins, likely used for huntng. Also sometimes thought to have been a reason for the expansion out of Africa, as apparently humans still didn't want to end up on the receiving end of 150 people throwing javelins.
The bow is itself a Johnny Come Lately at the scale of human existence. I wasn't referring to it as the first (thing that could technically be used as a) weapon, but rather as the thing which may have been a catalyst for human warfare. Humans had clubs, axes, spears, etc, before and seem to have mostly given other tribes a wide berth. Then suddenly they get bows and start painting a cave full of images of using it on other people.
I know that correlation does not mean causation, and, really, nobody can prove causation there. But it's a very compelling hypothesis, you have to admit.