The problem here is one of modern misinterpretation and misunderstanding of how combat actually works. Medieval soldiers didn't pair off and spar with each other. Nor did they really categorize themselves by weapon type. We like to call soldiers "spearmen" or "axemen" or whatever. Medieval sources are often very vague about what weapons particular kinds of soldiers might be using. In some cases, it's easy to tell that a crossbowman is a crossbowman and would fight in a group with other crossbowmen, but no medieval payroll divides troops up by whether they carried a spear or an axe. Talking about what weapon is "better" or more "effective" is a modern anachronism. From the medieval warrior's perspective, why be exclusive about weaponry? He can carry a spear in his hand and have a sword belted at his side to switch to when the situation warrants. Whether an individual knight carried a poleaxe instead of a lance, or a mace instead of a sword, was not what would decide the battle. Our sources speak to a wide variety of weaponry present on the battlefield, without much consideration to whether swords/axes/spears/hammers/picks/whatever were "better" in some abstract sense.

As a side note, I see many people commenting about swords as a "status" weapon that was too expensive for common soldiers to acquire. That may be true for the early middle ages, but by the high/late period, this was absolutely not the case. To be good with a sword took a fair amount of training, but that's the same for all military skills and medieval weapons. Pikemen must be drilled and disciplined to form an effective line, crossbowmen need to learn how to load, aim, and shoot efficiently, cavalrymen need to learn how to ride horses. Many youtube "experts" get very involved in these kinds of debates, and I don't want to dismiss them entirely. There's often some pretty good information that can be gleaned from the better ones. But overall, they have a very microscopic focus and don't seem to have much of a handle on any actual scholarship. Reducing medieval combat to rock paper scissors doesn't work and provides no useful understanding of combat.

This is towards the tail end of the middle age and the early modern period, but during the age of pike based warfare (Burgundian and Italian Wars especially), swordsmen were an important element after the pikes actually met; with plenty of armor and a shield, swordsmen could advance on the enemy pikes and get within their reach and dispatch them, disrupting their formation and allowing the allied pikemen to press forward and cause the enemy's line to collapse.

Additionally, something like a longsword is quite useful for combat between knights in plate armor; it's a very agile weapon, making it easy to parry an enemy's strikes and allowing you to take advantage the recovery time on weapons that might be better for actually hurting someone in full armor. It's very difficult to stop a mace or warhammer in mid swing, but the balance on a longsword is much closer to the hilt, giving you very good control over the blade, doubly so when halfswording. Killing someone in full plate was quite a lot of work, and involved a lot of grappling; a longsword gave you better leverage when it came to getting your opponent off his feet, whereupon you can draw your rondel and stick it in his armpit or his eye or his groin.

If I had to explain the success of the Roman gladius, I'd chalk it up to its superior flexibility compared to Greek style spear and aspis phalanxes; it would be much easier to take advantage of the disruption and gaps in enemy lines that followed the volleys of javelins if individual soldiers didn't need to march in lockstep with overlapping shields.

Swords for a lot of the Medieval period were a status symbol more than anything and only available to the higher classes. They showed the wealth to own one and also the fact that they could, and had trained since a young age to use one, something which anyone further down the classes wouldn't have had the time to do.

As to how they were used, often knight, would be fighting other knights who probably wouldn't be using spears themselves, if they were fighting each other. They would also be drawn after the very probable chance of losing the lance/spear on a cavalry charge. We know this as a very high amount of battlefield graves contain bodies who have been struck from above.