There's that (probably apocryphal) story about a Swiss militiaman who, upon being asked what the 500,000 Swiss soldiers would do if a million Germans invaded, answered 'shoot twice and go home'. I've always wondered why pre-gunpowder armies didn't do something similar.
Mathematically, it makes sense - if you have 5,000 soldiers and the enemy has 10,000, why not equip every soldier with a bow and shoot down the enemy army before contact? And if you outnumber the enemy, so much the better - more arrows for fewer targets! I'm thinking primarily of foot archers - I know mass horse archer armies have been common to parts of the world. I'm wondering why Harold's whole army at Hastings didn't fire a few arrows and head home, or why Alexander didn't conquer the world with tens of thousands of fast-firing archers instead of unwieldy phalanxes.
Economically it makes a sort of sense too - surely a weapon which is mainly wood was far cheaper and easier to produce than kitting someone out with a metal sword and a big wood and metal shield? And issuing bows surely does not prevent a soldier from putting the bow away and drawing a close combat weapon anyway - I'm picturing the painted image of the Immortals with their spears in their hands and bows on their backs.
As for why I think armies didn't depend on the bow, I would suggest bows are less effective than I'm giving them credit for. Perhaps the English longbow could whittle down a closing army, but most bows were smaller and weaker, or shields were bigger or armour thicker or horses faster, to make a difference before contact. Perhaps other projectile weapons like the pilum served the same purpose without the need for extensive training in two disciplines. And presumably bows and arrows are largely useless anywhere but open battlefields - no using them in cities or forests.
TL;DR - if you have thousands of guys, they could fire tens of thousands of arrows, and that seems like it would deal with most problems.
That is because bows weren't all that powerful by their own. The longbows used by the late medieval English were the exception - not the norm. The French considered their bows inferior to the crossbows, which can be seen in armourer proof marks. From the Angers armourers, 1448:
"... they will make white harnesses for men-at-arms fully proof tested, which is to say to crossbows with windlass, or at least half proven, which is defined to be from levered crossbows and shots from a bow, and however it strikes.The brigandiers [brigandine makers] will be held like this to make brigandines... the heaviest of up to 26 to 27lb, will be proof tested against crossbows with windlass, and stamped with 2 makers marks, and the lightest of 18 to 20lb, tested as above against a crossbow with a levered hook, stamped with 1 makers mark."
The bow seems to be considered the lesser threat here.
Going back to the question, the thing is that training a bowman takes time. Not everybody can just pick up a bow and expect to do well with it, it requires years upon years of training. The aforementioned English people trained from the age of 13 to be able to draw powerful bows. If you tried to equip people not used to bows with bows, all you would accomplish is a sub-par ranged force that would have little effect.
But now we get to the most important part - armour and protection. Usually the attacking army would be protected enough (either by armour or with the use of shields) to be able to advance under arrow fire. While the attackers may take losses during the advance once they reach the lines of the archers the archers are in deep trouble. This happened at Agincourt for example, even though the English had about 8000 archers the French were still able to make it to the front lines and the main fight happened between the dismounted French men-at-arms/knights and the dismounted English men-at-arms/knights (with the archers admittedly also joined in, but who were not as effective in the fighting as the knights).
Long story short, ranged weapons have seldom been enough to win battles of their own. It's the potential they have to disrupt enemy morale and formations thus giving their own side a better chance in the melee that makes them so useful in a medieval battle.
I commented on a very similar topic in a different subreddit (the discussion about prevalence of archers in Chinese armies until 19th century), where I addressed some of the ideas discussed in this thread (e.g. why the battle of Agincourt is not a proof that bows are exceptionally good weapon).