For a whole host of reasons. Most notably because archers weren't as important in battles as you think. Archers were great for any commander, as a screening force, as scouts, during sieges either as attacker or defender, archers have many uses. Pitched battle isn't one of them. It's remarkably hard to position archers in such a way that they are able to fire for a prolonged time without either being run down by enemy melee troops, being shredded by enemy skirmishers or are out of range/risk firing on your own men. It did happen now and then such as Agincourt, Crécy and Lechaion but these are the noteworthy exceptions not the rule.

Even if we disregard this. a war bow is a big, cumbersome yet surprisingly fragile thing that costs quite a bit to make. Say you give it to the regular spearmen and teach them to fire a few volleys, then what? Are they just going to drop their expensive bows in the mud en masse and charge? Carry them into melee? Either you end up using expensive bows as disposable equipment or you end up crippling your infantry.

You could of course in theory have a unit of melee troops trained to fire in loose formation, drop their bows, move forward, close formation and engage while auxiliary staff collect the discarded bows until after the battle but this seemes like a great deal of extra effort to go through for almost no added benefit.

In either case, this is a sidenote since archers and melee troops do not generally fill the same roles or occupy the same positions on a battlefield. The only notable exception are mounted archers from some cultures which could and did absolutely double as light cavalry, especially during pursuits.

Archery was a very specialised skill. We're not talking about a basic survival skill. As a military art, archery was not something that anyone could pick up and be effective at. In most cases, any soldier or warrior who used a bow in battle had to spend years to develop the skill, either from being born and raised in a culture where it is a survival tool (the native American nations, the steppe nomads of Asia and Europe), or cultivated through a culture and tradition (medieval England, China, etc.).

For bows to have any effect on the enemy, they need to be shot with sufficient accuracy and power. This is what the lifetime of training created: archers who had the physical capacity to shoot heavy war bows (anywhere from 70lbs to over 150lbs), with the stamina and control to shower enemies over 100m away, and shoot with "certain kill" confidence under 50m or shorter. The training involved specific conditioning of the body and correct technique to be taught, as documented in manuals such as Gao Ying's Ming military archery. Skeletons found in England show deformed bones through years of archery practice - where such training began an at early age so that future archers were taught to "lay into the bow" with the correct technique in order to pull bows of extremely high weight.

An untrained soldier would be physically unable to draw the bows used for war. Should they be given light bows, they would not have sufficient accuracy over sufficient distance, and any hits would not overcome armour. Archery was not about showering as many arrows as possible to cover a wide area. Quite the opposite: most sources and manuals place emphasis on aimed shots, particularly at closer distances where penetration was more likely and hits were almost guaranteed. "Point blank" for archers, depending on their bow, would often be around 50m, and trained archers were highly likely to hit their targets reliably at this distance. The closer, the better. Gao Ying even goes on to write that whatever distance an archer could ensure a hit, half the distance would be the effective combat distance - the message being that as long as you were not within reach of the enemy's weapon, you are at a good range to fight.

It was far more likely that an archer would be trained as light infantry. Skill with spear and sword is fairly easily taught, and archers were already conditioned to fight in hand-to-hand. The English archers, for example, were equipped with sword and buckler, and fought successfully in melee at Agincourt. Mongol horse archers were capable with sword and lance. The Japanese samurai were archers first, and were equally competent in spear and sword. The pattern is that you give your archers spears and swords to extend their use in battle, but you wouldn't give your spearmen bows.

You also have to consider the logistics of equipping an entire army with bows. The English might be the extreme, having mandated archery practice and stacked their armies with thousands of archers. The demand for wood for bowstaves and arrows necessitated a reliance on imports from Europe, specifically Spain and Italy. To equip every soldier with a bow would mean multiplying this demand by three to five times - Agincourt, for example, probably went through half a million arrows on the English side for their ~5000 archers. Then when you consider the composite bows popular in Eastern archery, which was far more laborious to make, you have a similar problem. There's a reason why spears were the most common weapon across all of pre-modern warfare: they were the most economical weapon to manufacture en masse to equip an army.

Then there's the practicality of even carrying the bow and arrows into battle. Already laden with spear, shield and perhaps a sword, requiring the carry of bows and arrows would be pushing video game levels of inventory management.

The role of infantry is not to engage the enemy at distance, but to carry the fight in hand-to-hand. Anything which detracts from their ability to do so would compromise their effectiveness on the battlefield. The solution to giving infantry a stand-off weapon was already present: the javelin. Cheaper to make, easier to use, and effective enough as a short-range throwing weapon.