The key idea to understand here is that not all combat is warfare.

Medieval violence can happen at many levels, just as they can today, ranging from bar fights to wars between armies. The sword is important because it can be useful in most points on this spectrum.

The 14th century Italian master of arms Fiore dei Liberi described the sword like this, in his book The Flower of Battle:

"I am the sword, deadly against all weapons. Neither spear, nor poleaxe, nor dagger can prevail against me. I can be used at long range or close range, or I can be held in the half sword grip and move to the Narrow Game. I can be used to take away the opponent’s sword, or move to grapple. My skill lies in breaking and binding. I am also skilled in covering and striking, with which I seek always to finish the fight. I will crush anyone who opposes me. I am of royal blood. I dispense justice, advance the cause of good and destroy evil. To those who learn my crossings I will grant great fame and renown in the art of armed fighting."

Here Fiore is talking about a sword that is commonly referred to as a longsword in the modern Historical European Martial Arts community. This is a sword that is primarily used in two hands, but light enough to be used in one hand too. Note all the different ways the sword can be used in that passage. The sword is long enough to have a chance against polearms like spears and poleaxes, but also flexible enough to be used at close quarters against other one-handed weapons or daggers. The sword can be used in one hand, two hands or as a short spear via the half-sword grip. The sword can act as a good cover that allows the wielder to move in close for grappling and disarming the opponent. (All images above are from the Pisani-Dossi version of The Flower of Battle) And perhaps most important of all, the sword can easily be carried at your hip. You might not always carry your spear with you, especially outside of war, but you can always carry a sword easily in everyday life.

The sword is the jack-of-all-trades weapon that is useful in all circumstances. Late medieval fighting treatises depict the sword being used without armour and with armour, on foot and on horseback, as the only weapon carried or as a backup weapon alongside another main weapon. When you are in sudden danger, when your spear has been broken, when you have nothing else, you will have your sword. And THAT is the advantage of the sword.