A sword is a sidearm, not a primary weapon. Once you drop your pollaxe or break your lance, you're gonna need something, and a three foot iron bar with every inch of it being designed to murder people is a good choice.
Swords were often used on people in civilian garb, or on people with little to no armour, where the long edge comes to a great effect. Also, as soon as something is exposed on your oponent, the entire length of the sword can be brought to bear.
But yeah, swords weren't a primary armament for dealing with armour, pole-weapons were, both amongst the rich and the poor. And yeah, there are going to be lightly or unarmoured soldiers on the field as well, at least well into the 14th century. Knights were armoured because they were rich. Those of lesser status who weren't supplied by their lord would need to get their armour themselves, and in a time where the shield was the primary defense, they were far more likely to have a helmet than a maille.
Also do take into account that you are not a profesional soldier who's used to murdering people with your sword, as such, tests by you may or may not be skewed.