Despite what that one duel in Game of Thrones tells you, a man in plate harness (or full mail protection) is not necessarily immobile or especially encumbered. Armor is a significant amount of weight, but modern soldiers also are encumbered with a fair amount of gear and can still fight very effectively (also consider the stiff uniforms and awkward gear of the 19th century soldier). A medieval man-at-arms, wearing plate harness, would be used to the weight of his equipment and likely had fought in it before. It's still going to impede your ability to move at a full sprint, or to see and breathe when you have your visor down, and it will be uncomfortable to wear for long periods of time. But it clearly did not immobilize soldiers or prevent them from effectively killing people, and the advantages of wearing a full harness were immense.

There is actually a great example of the protective value of armor in Season 1 of GoT when Jonah Mormont fights that one pseudo-Mongol/Hun/blend of steppe people stereotypes: https://www.youtube.com/watch?v=WRYM6B7CTs8. The combat in it is pretty goofy, but the conclusion of the fight is pretty accurate to what would happen if someone tried to slash through a breastplate with a sword: they accomplish nothing and the armored man will easily immobilize their weapon and dispatch them. Period sources, from the Crusades to the Hundred Years War, will describe men-at-arms being riddled with arrows without much damage due to the protection of their armor. It could be very hard to kill an armored man who is determined not to die, which is one reason why more casualties would be inflicted during a rout when the enemy was fleeing than during the main engagement. Going after the joints in someone's armor or outflanking and mobbing them wasn't a specialized tactic or only suitable for special units. That was what everyone in a fight, knights and men-at-arms included, was trying to do. Just about every soldier in a medieval army would have carried a dagger of some description in addition to their other weapons, so there would be no need to specifically arm them with one or to establish a specialized dagger unit.

That said, what you describe did sometimes take place, albeit in different circumstances. English archers in the Hundred Years War would be used as light infantry when they had expended their arrows or the fight came too close to shoot effectively. They were used to support a primary line of armored men-at-arms, either to reinforce gaps and weak points in the line or by outflanking enemies engaged with the men-at-arms. Several sources about the Battle of Agincourt mention the mobility of the archers as one of their advantages, but it must be remembered that Agincourt was fought on muddy terrain in wet conditions. In a stand-up fight of archers against men-at-arms, the archers would be at a distinct disadvantage, which the English (and all medieval armies) would often attempt to alleviate by the judicious use of defensive terrain or the construction of fortifications on the field. At the Battle of Fornovo, which was fought in and around a slippery riverbank, it was reported that the lightly armored French infantry, wielding one-handed swords, performed very well against the heavily armored Italian men-at-arms, wielding longer swords and polearms. This was, like Agincourt, more of a factor of the terrain and the immediate circumstances of the battle. Neither the French nor the English specifically made their troops wear lighter gear in order to take advantage of mobility. In fact, the exact opposite trend occurred as armor became cheaper through the 15th century: more and more "common" infantry would be equipped with at least some form of armor, even if it was only a helmet and breastplate.