I think Emax has a point. The bow is an ancient weapon and people of all social standings - at least the country-dwellers used it often and were probably adept at it. Even said peasants IF free men. He pointed out correctly the game they were allowed to hunt: rabbits, fowl, foxes to protect their chicken, squirrels and what not. Noble game like deer, boars and stuff are restricted to nobles by law, right. So unless they are un-free, people usually used bows to supplement their dishes, got food in winter and similar stuff. I remember from somewhere that there even were laws upon what fur commoners were allowed to wear. Do you really think those furs grew on trees (well, squirrel: yes, they do) or that there were specialized huntsmen everywhere who shot animals for fur for the masses?
About 95% were peasants and worked the fields but it ignorant to believe that plowing and sowing was all those people were able to do. They did carpenter's work when building structures and yes, they went on the hunt if they had spare time to shoot some meat for the cauldron over their fires.
Given that those people usually worked from dawn to dusk, did much work on foot and with manual tools and I believe while uneducated, they at least were generally stronger than nowadays city-dwellers. And they trained their skills from childhood on - maybe because there was not much else to do and had to grow up fast.
Lastly, I cannot believe, that nobles would disencourage their peasants from using bows. If they were free men, they were oblieged to serve in the army in their Lord's wars and I guess a nobleman is pragmatic enough to think his peasant is better alive than dead because who will pay his taxes otherwise. Contributing to the fight with less danger is either way a good thing and was frowned upon by nobility who could not accept that this kind of warfare was indeed more effective. Nobility was originally based on martial prowess. By not accepting archery as martial prowess you can keep your definition of nobility but I guess no noble but the most foolish was neglecting the power projection of archers.
From what I see is that we MIGHT need an encumberance system so units attacking or running or performing actions should lose stamina. I guess firing a bow once is not so much a big deal, if you are used to it. But doing so after forced march repeatedly? You cannot tell me, that cadence, accuracy and power of an archer would not suffer over time in a prolonged engagement. Same with exhausted melee fighters. 3 Javelins in the shield, plate armor and still running like Usain Bolt is very much as ridiculous as complaining about archers are powerful. Of course they are when you run straigth towards them. That is, why there are skirmishers to whittle down their numbers, flanking light cavalry or heavy infantry moving with a shield wall active (which requires discipline and shields large enough). Worked for the Romans.
Most armies have reserves for a reason to pull out men who are tired or wounded and keep fighting. In the game a wounded guy hits as hard as a healthy guy, runs as fast and swinging weapons does not encumber him. And armor works weird. Might we maybe see the root of the problem? If everybody behaves like Superman or Asterix on magic potion then of course ranged beats foot every time.
The problem is not that a bow is a good weapon per se - the game simply treats behavior of cobattants wrong. At least in campaign mode. In Multiplayer buff or nerf at your leisure - this is E-sports. But if you want to immerse into what it feels like fighting in and against medieval-style armies and troops, the game should behave 'realistically' with no need for balance at all.