Fun little fact, there's a lot of records of western colonists losing battles with local populations, because bows could out-fire, out-range and out-accuracy the musket rifles. We didn't switch to guns because they were better weapons at the time, but because you could train people to use them a hell of a lot faster than the comparative lifetime it takes to learn the sword and bow.
Yes but also no. It wasn't the musket's fault ;P
So one of the important issues regarding native archery was its arrows: Stone-tipped arrows were not as effective in warfare as they were in combat. Once they start importing iron from European merchants, this of course changes, as iron-tipped arrows gave them significant advantages when fighting against native rivals or colonists. However, this also brought them into contact with guns, and when given the choice between purchasing guns or sticking with archery, they picked guns. Evidently training was not an issue for native archers, as the training of a native archer started with his grandfather, but they nevertheless picked guns.
When native archers fought colonists using muskets, they often had a higher rate of fire, accuracy and range than the colonists.
When native musketmen fought colonists using muskets, they often had a higher rate of fire, accuracy and range than the colonists.
When native riflemen fought colonists using muskets, they often had a higher rate of fire, accuracy and range than the colonists.
Keep in mind, in Western Europe the bow went obsolete around the late 1500's, when the musket still remained inferior to the longbow. It had superior power, accuracy and rate of fire in the hands of a skilled archer, but it was simply easier to outfit and train larger masses of musket infantry who could accompany pikemen. The erosion of the available pool of skilled archers (as training starts with the grandfather!), the depletion of good yew with which to make longbows, gradually made it more and more apparent that the musket in spite of its deficiencies was going to replace it. By the 1700s the gulf between firearms and bows had been closed, multiple balls could be loaded in each shot which with longer barrels, solved the accuracy issue. And unlike arrows, lead balls could not be dodged or deflected by brush, and they were certainly very scary weapons. The invention of a rifled barrel tips the scales heavily in favour of the firearm.
The differences in the attitudes of the respective sides lay in the quality and doctrines of the respective soldiers and their equipment. The European colonists were largely agrarian peasantry, who hailed from countries where the governments kept them disarmed in order to keep them docile. They were not very well-practiced in shooting at marks, and when it came to warfare, its soldiers were trained to fire in volleys and close combat. This is why the European colonists still favoured the musket when the rifle existed besides it; the musket could fit a bayonet, the rifle at the time could not. They practiced for a conventional European war, which was largely useless against native skirmishers, who with the use of snow shoes, canoes, camouflage and effective tactics (rather resembling modern infantry fire team tactics), could inflict damage upon their enemy and force their withdrawal, or else themselves withdraw should the enemy charge or deploy cannons. The native Americans were skilled hunters, thus whether they used bow, musket or rifle (and they eagerly adopted the rifle, where the European colonial militaries were skeptical), they had years of experience before they even applied their skills to warfare. Where the European colonists were trained to maintain unit discipline and fire volleys at the enemy, native infantry were already skilled in the practice of shooting at individual targets. The consequence would be as long as confrontation with European forts and cannon were avoided, and mobility maintained, native infantry had nearly all superior advantages in combat.
The reason why they still lost in the end can be seen in the American civil war, where the side which had the superior marksmen lost to the side which had the superior industry and logistics. American forces had access to railways, cannons and steamers in a way which native forces could not replicate. And while native forces could exploit colonial rivalry to procure weapons, gunpowder and munitions from the French, British, Spanish or Dutch, when the USA begins flexing its arms and muscling away European influence - native ability to procure gunpowder is severely restricted. The invention and mass production of repeating rifles, and their adoption by the US military or its frontier colonists, in addition to the adoption of native tactics learned from native allies, led to material conditions which rendered ultimate victory unattainable.
Thus while the statement that guns replaced bows owing to the difference in training is true of Western Europe in the 1500s, it's not so true elsewhere in geography or time. In places like the steppes or China, they had sufficiently large pools of skilled archers that they only replace them in warfare in the turn of the 1900s, when Russia and the European maritime powers begin utilizing the first modern rifles and cannons. States like the Ottoman Empire are also cool examples, because they had successfully retained skilled archers over utilizing musketmen, until massed casualties in their naval campaigns of the 16th century wiped out too many of their archers for all of them to be replaced. Ignoring the technological innovations which had by then, made the musket the superior of the crossbow, there is another reason which perhaps contributes to the bow's replacement by both sides. It takes a strong and skilled man to use the kind of longbow which still conferred advantage over a musket. Yet in the hands of a weak man, or tired man, unskilled man, boy, peasant, hunter, sick man or injured man, a musket ball remains as deadly every time. And once you start training marksmen and snipers, you're better investing the large quantities of time you need to train a sizeable body of accurate infantry