If firearms "owned" archers, then why were both longbows and crossbows used as late as start of the 16th century? It wasn't because firearms were more expensive since the loading mechanisms for crossbows were more expensive than handcannons. However both firearms and crossbows were also used by armies who had longbowmen. One of either firearms, longbows or crossbows is most likely a little more effective than the others (though it's highly controversial which one is better),
but none of them "owned" the others. Assault rifles own swords and breechloaders own handcannons, but it's highly unlikely for weapons completely "own" another weapon from the same time.
The lingering of archery despite its being outclassed by firearms as a battlefield weapon is explained by a number of factors, (and actually, archery was around in many places well after the start of the 16th century).
1) Although firearms became the preferred weapon, many societies did not have the means to adopt them on anything like a wide basis. It was a long process to get the money, infrastructure and specialists to produce or acquire them in numbers and then also get a consistent supply of powder and shot. Even the Ottomans who had lots of resources could only consistently equip the Janissaries with them for a long time. But note how the incredibly effective and feared the Janissaries were!
2) Traditions. People were used to archery. The knowledge of archery was everywhere, there was a huge inventory of archery weapons around, and there were established systems, even whole societies such as those employing horse archery, Tartars, Mongols, etc. build around it. That just can't change fast. Furthermore, communication and education were not as developed or as widespread as now. Change took time, especially then (even when there was a big motivation for it) and no one was just going to throw away an inventory of weapons. But clearly those in a position to adopt firearms did so as quickly as they were able to.
2a) The implications of gunpowder were felt not only on the battlefield. Adopting gunpowder meant that certain industries, professions and social classes were expanded and increased in importance. More money had to be raised and in different ways, etc. There was resistance to those changes from those who would not benefit or would lose importance. It was a doomed resistance but it was there.
3) There were circumstances in which using firearms was problematic. For example, using match-locks on horseback. Firearms were not used widely and effectively from horseback until the development of the wheel-lock in the early 1500s. The wheel-lock quickly became a very high demand item but it was complex and costly to produce, certainly more difficult to produce than a crossbow.
Match-locks also had problems in the rain or other damp conditions such as at sea. Those could not be heavily reduced until the flintlock came around.