Well, swords can stab as well as slash, so it’s not like they weren’t useful…although historians of arms and armor note that from the 10th century onwards, you start to see sword blades becoming more slender, suggesting a switch in emphasis from slashing to stabbing. (Hence the development of half-swording and the Deutsche Schule, which hardly ever using slashing attacks.)
Knights would certainly have trained in the use of the mace, morning star, handaxe, warhammer, etc. and would have used them all the time, so it’s not like those weapons were neglected.
So rather than the sword being the default, I would call it instead the symbolic weapon. Axes, maces, warhammers – these are civilian (i.e, peasant) tools pressed into service in time of war, whereas a sword is designed only for war. Thus the sword becomes the badge of office of the knight whose profession is war – and all of the symbolism around knighthood (dubbing, swearing fealty, etc.) revolves around the sword.
And yes, lest we miss the obvious, the fact that the development of knighthood all took place in an intensely Christian context – especially when the Crusades start up and you get the development of chivalric literature – it didn’t hurt that the knight’s sword is shaped like a cross.