It was actually double mail that allowed fighters to stop using shields and focus on two handed weapons; with single mail, no shield= dead fighter. Two handed weapons then stimulated armour makers to produce plate to counter the increased power of weapons like the longsword (which historically was always used double handed). Ironically, the weapon most plate-armour wearers would fear is the rondel dagger; a dagger with a very stiff blade with a triangular cross section. The best tactic, advocated in medieval fight-books of the time, was to trip the plate wearer with either grappling or a polearm or longsword (you reversed the longsword and used the hilt to hook behind the knee or ankle) and then, once they were on the floor, drive a rondel through a joint or the visor. Having said this, once you had a knight down, it was time to think about ransom not killing; such were the economics of the day.
Yet double mail and plate were not common equipment on any battlefield and most people would not have had access to them; they were too expensive. Also, knights in plate frequently used shields because a lance can penetrate full plate if your opponent is galloping towards you on a horse. The shield was an essential part of defence against this. So sword and shield was very important and the bottom line is most of a trained fighter's defence came from his shield, not from armour, unless he had plate. Parrying is another misconception; most medieval blades were not used for parrying much; you blocked with a shield or buckler and then this opened your opponent to a counter with your sword.
The sword and shield combination was also essential for lightly armoured soldiers for one very important reason; if you don't have metal gauntlets then you MUST have a bucker that constantly covers your sword hand or else it will be cut off as soon as you attack with your sword; the defender simply targets your exposed arm/hand. There is a special sword and buckler style that was used from 11th to 18th century where the buckler hand hovers over the sword hand except when shield bashing and during some guards and is illustrated in a document called I33 from the Tower of London; odd that the buckler was the MOST common shield to bash with and yet you can't do it in 3E D&D.
You could make a really good combat system from historical martial fight-book moves and it wouldn't feel that different to 3E combat in many ways; just the emphasis on sword and board.