Not to take away from either of the other answers...
Plate armor is NOT full body protection
Having worked in ren faires, I've seen a fair share of "full armor reenactments". To which, there are many areas, even in what would be considered full plate armor, that give only the minimum of protection. Mostly around the joins (like elbows and knees), the groin, and other areas depending on the manufacturing. People need to be able to move. Here is an example of armor with a few exposed areas.
It is not hard to imagine that a trained swordsman could target these areas and get a slashing wound to their opponent.
Real longswords are bludgeoning and slashing and piercing
Many swords (that D&D would refer to as "longsword") actually do all three types of damage.
For instance, a number of blades have a Ricasso, or an unsharpened area of the blade, general for gripping with a spare hand, meaning they were only sharp towards the tip.
Combat would entail first bludgeoning your foe to create chinks in the armor. This would would likely take the form turning the sword around, grabbing by the ricasso, and hitting with the pommel/hilt like a hammer. They could also use this for prying open/off armor.
Now that there are more exposed areas, the fighter can either swing and slash at the target, or jab straight in and pierce the target.
But having to keep track of all that would bog down a game. So WotC simplified to (most) everything having a single damage type.
As Thomas pointed out, changing the type of damage every round would bog down the combat considerably. For instance, how many bludgeons would need to hit a creature in plate armor until you could start doings slashing/piercing?
It's just not worth the hassle. Generally, the game treats bludgeoning, slashing, and piercing damage in the same bucket. Barbarian Rage resists all three, the Heavy Armor Master feat treats all three the same, etc.
The only time such a difference would matter would be on things like the black pudding which is immune to slashing, but bludgeoning and (oddly enough) piercing work fine. Or objects, may only be susceptible to one or two damage types.
But those are the times to use ingenuity and not make up rules on how to change damage type.