We do have fair amount of literature from late 16th century England comparing the relative merits of firearms and longbows at least. In general proponents tended to praise them for their incredible range and power compared to contemporary weapons, especially heavier weapons such as the musket.

Probably one of the biggest proponents of firearms from the period was Humphrey Barwick, an English mercenary who after spending his career fighting wars on the continent wrote a discourse essentially outlining why he thought firearms were superior to bows and crossbows in pretty much every conceivable way. He even had answers for the slow rate of fire (according to him skilled musketeers should know how to elevate their weapons and start shooting volleys at up to 480 paces, allowing them to get more shots in against a closing enemy than archers could. Then at close range the muskets and arquebuses could be loaded with multiple bullets at a time, turning them into shotguns).

As for crossbows, while a steel bodkin is more efficient at penetrating than a lead ball for the most part the energy difference was insurmountable, especially with armor growing thicker and thicker. It takes an extremely powerful crossbow to shoot a bolt with more than 200 joules of kinetic energy, many 16th century firearms on the other hand could easily shoot a bullet with up to 3000 J. The largest musket from the Graz tests had a muzzle energy of almost 7000 J. Also, while the relative accuracy of the weapons might see some debate there is definitely something positive to be said about the weapon that has the highest velocity, since it makes it easier to hit moving targets and estimating range is far less critical.

So, I had written out a long post about the pros and cons of gunpowder weaponry vs. bows and crossbows. /u/Rittmeister has an excellent explanation of those developments here, though there are dozens of such threads throughout this sub. So rather than regurgitate yet another one of "those" posts, Id like to take my answer in a slightly different direction.

If you really wanted to pin the demise of the arrow in favor of the bullet, The Battle of Pavia is probably as good a point as any to look. Specifically, at Pavia in Italy, thousands of Swiss mercenaries fought for the French cause. They were opposed by German mercenaries called the Landsknecht. For many reasons, the French army at Pavia was absolutely devastated by the German armies. However, one major cause of the defeat was in the equipment between the Swiss and the Germans. The Swiss favored the old style of warfare, and failed to incorporate many guns into their formations. The Landsknecht, on the other hand, embraced the new technology. Following Pavia, the Swiss star fell and was replaced by the German. Similarly, as armies reorganized, the did so around the successful German model, rather than the disgraced Swiss model.

So, lets come back to your original question. I would argue that bows and crossbows were not as effective as many commonly believed. Our friend Rittmeister discusses the hows and whys of that. But Pavia also demonstrated clearly, that when it came to the "Push of Pike" the more armies featured gunpowder weapons, the more successful they were. And in the simplest terms, armies (and especially mercenary companies) across Europe simply wanted to have the most effective and successful weapons around. Once the Swiss, and by extension the older style of warfare, were decisively defeated, European armies flocked to the new style.

Bows greatly surpassed early guns in fire rate. However crossbows that were capable of piercing plate armor were actually slower to load than an arquebus. The arquebus is also superior in armor penetration to both of these weapons.

An arquebus requires little training time compared to the bow, (Notably the English longbow) which required long training times to become effective in accuracy as well as being able to draw back the bowstring.

It is true that arquebuses were inaccurate but this was supplemented by the tactic of volley fire.

Despite all of this, keep in mind that these weapons had a lengthy period of overlap.