I think you're mistaking the phenomenon of people simply not living up their ideals with the creation of a myth. Chivalry was something that knights were talking about from the middle of the 12th century onward, though exploded in popularity to the point of cultural dominance c.1200. People who embodied the ideals of chivalric ideology - often called the prudhomme (literally "prudent man" or "wise man" in Old French) - were exceptionally rare. There are only about a dozen people widely agreed to be one, and that's if we're feeling generous. It was not a myth, just a high bar.

Three things seem to have come together in the middle of the 12th century to create chivalry: an identity crisis among knights, the repopularisation of classical philosophy in Europe, and the development of tournaments as a social sport.

In the 12th century, knighthood was in a bad place. One problem was that knights were worried by the growing body of free peasants and professional mercenaries who could do their job in war. Financially speaking, what made a knight was the ability to afford the best equipment and leisure time to hone their skills, and there were an increasing number of non-noblemen who could do that job. A knight simply being someone who could fight as heavy cavalry was no longer a sufficient marker of identity for knights. The other, bigger problem was that knights were generally regarded as bad people. The influential theologian Bernard of Clairvaux had written of knighthood as "a form of banditry" and the dismal failure of the Second Crusade in the 1140s was attributed in part to the sins of knights. Furthermore, England had just come out of a bloody civil war between King Stephen and Empress Matilda over the rulership of England. During this war, sometimes called "The Anarchy" by historians, some noblemen went rogue and pillaged parts of the English countryside while knights frustrated at a lack of pay or victory took out their anger on local people. The successor and ultimate victor of this civil war was Henry II, Matilda's son, and he was obsessed with preventing such a civil war from happening again by centralising power around the monarchy. With the twin catastrophes of The Anarchy and the Second Crusade showing knights at their least useful, and a growing class of freemen able to do their job, they needed to get their act together and show that they were worthy of being the elite.

This coincided with renewed interest in classical virtue ethics and classical literature more generally. The first hint of chivalry as a developing ideology comes from an English philosopher called John of Salisbury in his gargantuan treatise Policraticus, written in the 1150s. John was trying to work out how to stop The Anarchy from happening again, but was critical of many of Henry II's reforms. John's philosophy is rooted in political realism, and he points out that all the laws and customs and powers in the world won't matter if the people wielding those powers suck. If anything, centralising power just makes it worse by potentially centralising that power in the hands of a bad person - a prophetic statement considering the problems Henry II's reforms caused under King John and Henry III. His solution was to teach classical virtue ethics, mainly Aristotle, Cicero, and Seneca, to young noblemen and courtiers. He wasn't the only person with this idea, and many of the early chivalric tales were explicitly about revitalising classical virtue ethics. If the idea of the prudhomme as a noble leader enlightened by the practise of virtue sounds a bit familiar, it's because that's a medieval version of Plato's philosopher-king. It is also worth noting that two commonly agreed chivalric virtues, largesse (gift giving) and clementia (mercy), were directly based on Seneca's two treatises De Beneficiis and De Clementia, which were produced in a combined edition popular with noblemen. With many noblemen doing some soul searching, and a resurgence in classical books claiming to have answers, it is no wonder that many noblemen began reading classical philosophy and particularly virtue ethics. In time, almost every feature of classical virtue ethics would find its way into chivalry and some historians have gone as far as to call chivalry a "neo-classical" ideology.

But a few knights reading Seneca wouldn't have given rise to a new model of knighthood. It needed to spread. Fortunately, the tournament was taking off as a social sport for the upper class. The earliest tournaments were just war games in times of peace, but when knights realised how fun they were they became more of a sport with rules and regulations. This gave knights much more of a community, and a place to exchange ideas. In the day the teams would fight, and in the evening it was time to eat and drink and chat into the night. One of the social events that went with tournaments was philosophical discussions, whether through the performance of political songs or just readings of classical texts. Popular culture tends to have this image of knights as rather brutish and unrefined, but a lot of them were nerds. We can even track intellectual trends among the tournament reading groups, such as a craze for Cato the Younger (or, more accurately, texts that had been falsely attributed to him) in the 1210s. These trends tend to line up with when some aspects of chivalry became popular, such as the incorporation of Stoic ideas in the 1210s and talk of "moderation" as a virtue in the mid-13th century as parts of Aristotle were translated into Latin.

So if we have to pinpoint when chivalry became a thing, and when its ideas were being discussed most widely, I'd probably say the 1180s-1230s. In the 1250s we start to get the nostalgic "things were better in the old days" stuff like Ulrich von Lichtenstein's Frauenbuch, and whilst this kind of nostalgia is ever present, interest in classical virtue ethics does seem to be less prominent in chivalric literature, though chivalry still had a couple of hundred years to go. Then again, even John of Salisbury writing in the 1150s complains about knights being virtuous just because of societal pressure rather than because they've actually read Seneca and understood it, so perhaps it was always like that. Even so, it would be a mistake to think of chivalry as a myth. Instead, it was a social movement among the warrior-elite of medieval Europe encouraging the practise of virtue ethics with a very high bar for success, with most of its intellectual development taking place in the late 12th and early 13th centuries.