There's a couple dimensions to this question. One, mail was already a pretty good defense even before plate armor. It's not that mail was bad and therefore plate harnesses had to be invented. In fact, plate harnesses seem to have evolved from supplementary defenses worn on top of mail. The development of better protection later doesn't mean that earlier forms were necessarily insufficient. Even when plate armor started to become more common, plenty of soldiers still wore mail and mail was often used to cover gaps in early harnesses (such as in the armpits). Keep in mind also that not every soldier in the entire army would necessarily be equipped in the heaviest gear.

Two, it's not necessarily the case that English longbowmen were particularly destructive against plate armor or that Saracen archery during the crusades was useless. These are two very different contexts and cultures, but they share a principle of battlefield archery: the purpose of archery in combat is not necessarily to kill the enemy outright. Certainly death and injury will occur while volleys of arrows are raining down, but the archers do not have any expectation that every arrow sent into the air will necessarily kill or disable an enemy. They're shooting to harass the enemy, to agitate them, and to disrupt their movements. Even when missile bombardment is producing little in the way of direct casualties, it is still psychologically draining to be under fire.

Three, just because arrows cannot penetrate heavy armor directly does not make them entirely unable to cause damage even amongst the ranks of heavily armored soldiers. Despite their reputation in pop culture as anti-armor weapons, English longbows are often described as being somewhat ineffective against plate armor in period sources. Instead, arrows did damage by the aforementioned psychological impact, but also by hitting unprotected or worse-protected spots in an opponent's armor. Remember, archers firing in large volleys are not taking aim at individual opponents. They are dumping a high volume of arrows into a group of enemies and hoping that a few will hit in an elbow, an armpit, a horse's eye/leg/flank, or through the visor of someone's helmet. Thus, picking terrain and setting up the battlefield to ensure that the French spent as much time as possible getting hit by arrows was vital to English success in field battles. When the English were able to pick their ground and establish field fortifications that would delay an enemy advance, they had a good chance of victory.

The Saracen use of archery to harass Crusader armies was not entirely the same as English archers in the Hundred Years War, but they understood the same principle. Arrows were not expected to kill with every shot. Instead, by constantly harassing Crusader columns, they could physically and psychologically wear down the Europeans, which theoretically would weaken them enough to make a direct confrontation more viable. Horse archers were far more prevalent in Muslim armies of the period than they were in western European warfare. The advantage of horse archers was that they combined the harassment effect of regular archers with a huge degree of mobility. Horse archers could effectively shadow a Crusader force, probe for weaknesses, and hopefully draw off small bodies of over-eager Crusaders into ambushes where they could be surrounded and neutralized. They were not supposed to win the entire fight just by lobbing arrows into marching columns. I hope that answers at least part of your question, although perhaps /u/Valkine or /u/warritter would also like to weigh in.