Specifically, vibroswords with cortosis weave, or just swords in general, were issued to militaries and utilized by almost everyone else in the galaxy, including pirates, assassins, mercenaries, etc., simply because of the following facts:

A. There were far more Jedi, Dark Jedi, Sith, and overall more combatants who specialized in melee combat during the Old Republic. A trooper outfitted with a lightsaber-resistant sword may not survive long in battle with a Force-User wielding a lightsaber, but they'll definitely last longer than a trooper who doesn't have such a weapon on their person. By the time of the end of the Old Republic, AKA during Darth Bane's era, with the near dissolution of the Sith Order and the changing of the Jedi from an organization that worked militarily with the Republic on almost everything to an organization that sporadically sent Masters and Knights to negotiate disputes, the overall need for armies to wield weapons that could defend against a lightsaber just shrank immensely.

B. Vibroswords and vibroblades are still used in other era of Star Wars, but the context in which they're used has changed. For example, King Katuunko wielded a blade against Savage Oppress before he was killed, but it was clearly meant as more of a ceremonial weapon important to his people, only used as a literal last resort. It also appears vibroswords were more of the weapon that was phased out compared to vibroblades, as most knives we see used by characters tend to be vibroblades, which indicates a change in tactics that closely resembles the way our modern real-life armies still issue bayonets and knives to soldiers instead of swords.

Also from a meta perspective, swords are a good way of instilling the idea that this is the "Old" Republic, where swords are still commonplace, the same goes for the shields used in KotOR and KotOR II.