I think a sword is kind of just a good all-arounder. They're fairly fast and balanced and streamlined and have decent reach, while providing a lot of ways to attack. There's also a good defensive element to them, as they allow for either using a one-hander with a shield or using a two-hander to block enemy attacks. Two skilled enemies facing off with swords can create an epic battle, and that infuses more of a sense of "honor", facing your enemy in close combat and relying on skill more than brute strength.
The thing with spears (pikes, halberds, etc.) is they're more suited for creating a defensive line as opposed to open combat. A long stick with a sharp thing at the end is best for having a bunch of unskilled guys stand in a row and letting the charging enemy impale themselves on it. That, or you have a short, light spear that would be more mobile but would essentially be a sword made mostly of breakable wood with only a very small area of it that's actually dangerous. Also, if I'm holding a spear, it gives a cushion of space between me and my enemy, but once they get closer to me than the tip of my spear, there's not much I can do besides hit them with the shaft.
Axes and other large, heavy swinging weapons are also pretty limited. I swing, then I have to lift back up, then swing again, hoping I smash my enemy to bits the first time. There's not mush finesse involved, which makes them seem more like the weapon for some brutish enemies rather than the noble hero.
All of this is based on thinking about actual, "realistic" properties of weapons (at least what I've observed about them) as opposed to the mythical "sword-the-size-of-a-car, fighting-in-the-air-for-45-minutes" combat properties of some games.
