The short answer is that it is basically nonsense, but to fully untangle why that is takes some doing. There’s a lot of pre-existing assumptions wrapped up in this myth and fully addressing all of them is somewhat beyond me. I’m going to start with the area I’m most comfortable with, the technology, and then I’ll try and explain some of the more complicated socio-political aspects of it later, but just bear in mind that being a historian of medieval nobility is a specialty in and of itself so to fully explain those aspects would require the recruitment of at least one more historian into this answer.

As you note at the end of your answer, this is also a popular story about the crossbow.* The version of the story that I’ve heard the most often is that the Catholic Church banned the crossbow at the Second Lateran Council in 1139 because the crossbow was upsetting the power balance in Europe by allowing peasants to kill nobles. To get the partial truth out of the way, the Second Lateran did ban the use of bows and crossbows in inter-Christian warfare, among many other things, but the ban had no discernible impact on medieval warfare. As a point of reference, Second Lateran also banned jousting, tournaments, and fighting battles on most days of the week, none of which were followed.

A surprisingly relevant aspect of Second Lateran was the ban on warfare during certain holy feasts, like Advent or Easter. This was based on the Truce of God movement, which was a movement initially in France that attempted to restrict when warfare could occur. The Truce of God was closely affiliated with the Peace of God, which attempted to get nobles and kings to agree to sparing peasants, women, and clergy from the ravages of war. These movements were quite influential, although never wholly successful they would have significant impact on the development of the Crusades as an attempt to direct Christian warfare and violence beyond Christendom, and they underpin an important aspect of the Catholic Church at the time. The individuals involved in Second Lateran (it’s not wholly fair to say “The Church”, the clergy were not a monolith) were more interested in restricting violence against peasants than against nobles.

Another important factor to consider is the likelihood that a peasant would even have a crossbow in the first place. In medieval warfare, the crossbowmen tended to be an elite soldier – not of the status of a man-at-arms but higher than that of your standard infantry. A big factor behind this was that the equipment a crossbowmen used was more expensive and harder to get than that of a spearmen – soldiers’ rates of pay were often tied very closely to the cost of their equipment. The peasantry of medieval Europe was not awash with crossbows, instead it was a tool of elite warfare. While King Richard I was eventually killed by a crossbow – one of very few kings to be – he was also a major proponent of the weapon, he even personally shot one at the Siege of Acre and the relief of Jaffa during the Third Crusade. The European nobility loved crossbows, that’s why you see them in so many medieval battles and sieges. As a last point, if a lone peasant were to find themselves in a field facing off against a duke in full armour, they’d have time for about one shot before the cavalryman split their head open – crossbows were most effective deployed in formations supported by other infantry, they couldn’t stand toe to toe with a mounted man-at-arms.

So what about gunpowder? It is fair to say that gunpowder weapons changed the face of warfare, but it is often vastly overstated how quickly that happened. To somewhat oversimplify the subject, you could point to two major points in late medieval and early modern history when gunpowder weapons radically changed warfare. The first is in the mid-fifteenth century when the effective adoption of gunpowder artillery radically altered the balances of siege warfare. The exact extent of how effective this was has been hotly debated, but certainly the success of the Bureau brothers in the service of King Charles VII in driving the English from France and winning the Hundred Years War points to something being different. However, technology quickly adapted and by the end of the century the balance was already being restored – with the invention of the star fort in the sixteenth century sieges were once again prolonged slogs, they were just also much much louder and deadlier. Handheld guns were present in the later Middle Ages, but by and large medieval guns were artillery and meant for shooting at fortified walls or occasionally masses of troops rather than individual targets.

The next change was during the early to mid-sixteenth century when arquebuses began to dominate Europe’s battlefields. Arguably the most significant change the arquebus brought was making warfare way more deadly, an arquebus could punch through armour and kill at a much greater range than a bow or a crossbow ever could. Still, crossbows and bows lingered on – crossbows were definitely still used at the Siege of Malta in 1565 and the English monarchy only retired the longbow in the 1590s and that decision was not without controversy (whether that controversy was the result of mistaken nostalgia or realistic tactical insight is a debate for another time).

Here’s the thing, though, gunpowder weaponry in Europe is documented from at least 1326, and probably has its origins even earlier. Certainly, we know Europeans were experimenting with gunpowder in the 13th century. Guns were probably used by the English at Crécy in 1346, probably to little effect, and pop up in medieval warfare over the following century. Guns had centuries of history in place before we see them begin to really shake up European warfare.

One thing about this myth that always bothered me is that its very obviously rooted in people seeing eighteenth or nineteenth century reenactors with their muzzle loading muskets and pre-packaged cartridges. Sure you can teach someone to shoot a Brown Bess musket with reasonable competence in a couple of weeks, but that’s a technology with centuries of history behind it that has made it that way. The very earliest of medieval gunners had to mix their own gunpowder in the field. Even after the invention of corning which allowed the creation of gunpowder cakes which could be crumbled into pieces for shot, you still had to know how much powder to put into your gun to get a good shot without accidentally blowing yourself up (the latter being a real concern for much of the Middle Ages). Medieval gunners were highly experienced and capable people, often reasonably well paid.

To top it off, gunpowder was ludicrously expensive for most of the Middle Ages. Gunpowder requires three ingredients: charcoal, sulfur, and saltpeter. Charcoal was readily available in the Middle Ages, and sulfur could be easily mined in any of Europe’s volcanic regions (Iceland and Sicily being the most popular and prolific). Saltpeter, though, was much trickier to get – for most of the Middle Ages it either had to be imported from Asia (often from India) or discovered in the wild by pure chance. Sometime in the late-fourteenth century some parts of Germany discovered how to effectively produce it in pits of decaying vegetation, but they carefully guarded that secret for most of the Middle Ages. England was still trying to master making their own saltpeter during the Tudor period. This shortage meant that saltpeter could easily cost more than the guns it was shot from, which also handily limited who could afford it. For much of the Middle Ages, being able to effectively use your guns meant having deep pockets. Even if a fourteenth century peasant knew how to shoot a gun odds are they couldn’t afford the gunpowder to actually shoot it with.