From what I know the matchlock was the earliest and most produced gunpowder weapon, but why? From what I've read trying to answer this question it sounds like the matchlock was completely inferior to the bow. The matchlock took a very long time to reload, from what I've read 3 to 5 minutes, so a bow is definately going to have a larger rate of fire. If Calvary were to charge a matchlock unit wouldn't have enough time to fire a 2nd volley. The only positive thing I've heard is that, the sound is very loud that it has a large demoralizing effect, but as guN's were normalized that became irrelevant.
Guns can't shoot over terrain like bows, and guns are more expensive to produce, matchlocks. I'm not saying that guns are inferior to the bow, but it makes no sense how the original gun became so popular allowing for improvements, even though it was horribly inferior to the bow. Oh and matchlocks were much harder to fire on horseback, let alone reload. Why would any army at the time pick up a matchlock? Now I know there's going to be a point of how matchlocks could penetrate into armor, and I find that questionable. I believe that the Hojo clan during Japan's civil war, used thick wooden shields to protect their samurai before charging, and I saw some wooden shields used in a Korean movie against matchlocks.
Edit: When I say that I'm not trying to say that guns are inferior to the bow, I meamt that modern day weapons are definately better for killing and waging war than a bow, but the gun In it's youth was terrible and not cost effective.
You're only considering tactics, not organisation. You're quite right that firearms weren't clearly superior weapons in battle (this is why the idea that early colonialism was facilitated solely by Europeans' use of guns is nonsense). I don't think there's significant evidence that guns had any particular demoralisaing effect, since they became so widespread so quickly that any soldier with experience would be as used to firearms as they would be with any terrifying aspect of battle. But it was possible to train an arquebusier in a few weeks, compared to years for a skilled bowman. Gunsmiths sprang up relatively quickly, which helped to reduce the cost of the weapons, and it's not like it was cheap or easy to manufacture good bows either. I'm not sure your point about bows being easier to use from horseback is relevant since Europe didn't have a culture of horse archery anyway; cavalry did begin to use pistols from horseback, and where it did exist horse archery continued for centuries, even in countries like the Ottoman Empire which made extensive use of firearms.
The advantage of easy training was only compounded by broader changes in military organisation. Firstly, armies were becoming larger, and the monopoly of the nobility over the means of warfare was ending. This obviously made it preferable to train troops more quickly, especially with weapons that could be used effectively by those without any military background. A further ramification of this expansion was new forms of battlefield command; armies began to use precise drill to make up for the troops' lack of skill, and employ them in large, easily-controlled formations (like the Spanish tercio). This was mainly a solution to the problem of using pikes, which had become the pre-eminent infantry weapon in the hands of Swiss and German mercenaries but could easily be mishandled when given to unskilled levies--unless they were drilled to move in precise formation according to a set of pre-arranged steps. (See Thomas Arnold's The Renaissance at War for more on this). Matchlocks lent themselves to such a system since they were at their most effective when the gunners were drilled to fire massed volleys. It was also natural to adopt firearms since slow blocks of pikemen made ideal targets; while combining pikemen and gunners gave the formation some form of ranged capability while the pikemen could protect the gunners from enemy cavalry. This was so effective that in the 16th century cavalry abandoned their charges and adopted caracole tactics, in which they'd fire pistols at the enemy without actually engaging in melee, and it was only in the 18th century with the replacement of the pike by the bayonet that cavalry would return to shock tactics.
Additionally, armies were symbols of prestige as well as practical organisations, and modern technology played a part in this. Charles VIII of France gained renown for creating a siege train of modern artillery even though when he used this in Italy in 1494-5 most of the sieges were still decided by methods which would have been familiar to medieval soldiers. While not as potent a symbol as cannon, matchlocks were still cutting-edge technology.
First off, you are varying between the term "gun" and matchlock. A matchlock is a type of gun, but it is not the same as all early guns. Improvements are made on all sorts of things that later become popular because of those improvements. There were improvements made on the matchlock that improved upon all of the flaws you mention.
Matchlock reload time would have been nowhere near 3-5 minutes if you are referring to around the 15th century, however. It would be more like 30 seconds. And a range of 30 yards. They were also used in multiple volleys so that you would actually have a round going off every 15 seconds.
Bows simply could not have the range, accuracy, and deadly effect of matchlocks and following designs. The price of a weapon certainly would not affect its use if it is superior. Matchlocks had disadvantages that could be exploited, which still made them a vulnerable choice of weaponry. Native Americans, for instance, did not begin truly adopting to firearms in mass until the flintlock.
Cavalry were still using swords up until WW1. There were times for guns and times for swords. Cavalry are devastating to any type of unprepared enemy, but matchlocks could certainly get more than 2 rounds off before they would be reached by approaching cavalry. Horses would also be very easy targets, and once you shoot the horses, you end the cavalry charge.
The flaws of guns certainly did not outweigh the benefits as improvements were weighed. Initially there were instances of bows being used very effectively against matchlocks. The vulnerability of relying on gun powder and a "match" could also be exploited in close quarters. In large numbers and in open battle, however, the matchlock as very effective.
The idea of the Hojo Clan facing matchlocks is unlikely given that their reign ended before matchlocks were likely invented. I cannot find any reference to this happening. And "a Korean movie" is too vague. There were certainly attempts in history to stop the penetration of every type of ammo. I doubt thick wooden shields would be very ideal though.
http://www.nps.gov/jame/learn/historyculture/history-of-armour-and-weapons-relevant-to-jamestown.htm
I can only answer for Western European history but the simplest single factor is that musketry requires less training than archery. Your post also has some assumptions in favour of bows which aren't true
Source: Mostly general knowledge but checked with The Military Revolution in Sixteenth-Century Europe By David Eltis