So what you're saying is that if kids were raised such that winning was the most important thing, then failing at a game would have had them a lot more relaxed than if they were taught that they can just enjoy the experience of the game. Sounds rather counter-intuitive.
No. What I'm saying is that teaching kids that participation alone is worthy of reward (I.e. getting a trophy even if you lose) is detrimental to their development because it does not teach them how to properly deal with failure and inequality. There is nothing counterintuitive about it, as I will explain.
The activities that children participate in as they develop, as well as how they learn to interpret their participation in those activities, establish cognitive traits that become hard-wired over time. Excluding genetic factors, this conditioning eventually leads to how the adult brain interprets the world. Thus, how children experience their world has long-term implications. (I'm not going to cite any studies on this because it's all psych 101.)
By rewarding kids for participation rather than their degree of success in a competitive activity, they are necessarily deprived of the full experience of failure. While they may technically lose a game, they are still rewarded for their performance – fostering the idea that you are still a "winner" even if you lose, because you "gave it your best" or "had fun." Although these two points are important values to instill in children, elevating them to the same status as winning by giving them rewards prevents them from having to perform the level of self-analysis necessary to effectively respond to failure. They aren't forced to analyze their mistakes, they aren't forced to take responsibility for those mistakes, and they are enabled to ignore real inequalities. In essence, they do not have to overcome internal or external obstacles to achieve their goals. They are "winners", regardless.
Nature – and therefore everything in civilization – only rewards results. Simply showing up and "giving your best" will not cut it in most situations. Consequently, people fail at their endeavors. They may participate in the larger ecosystem, but that participation is never rewarded for itself; it's rewarded for its results.
The "everybody wins" value system sets kids up poorly for this environment. Instead of responding to failure with self-analysis, they look to the external worlds as the primary reason for the absence of personal achievement. Thus, young adults are conditioned to conclude the following:
Ex.1: I have a degree, so I should be given a well-paying job.
Ex.2: My job was outsourced because workers in [country] will work for slave wages.
Ex.3: If two people work for 8 hours, they should receive the same compensation.
Ex.4: I haven't had a successful relationship because I haven't found the "right" person.
Each of these arguments, which are synonymous with my generation, all place the blame for failure and loss on everyone but the individual in question, because participation, or effort, (rather than the quality thereof), is believed to be the determinant of their merit: (1) I went to school, so I deserve a well-paying job; (2) my labor is worth more than my employer is willing to pay; (3) all workers are of equal value; (4) the failure of my relationships is not my fault.
This reasoning, nurtured by the "everybody wins" culture, leads them to ultimately conclude that their efforts aren't paying dividends because the system is faulty. Consequently, they do not investigate and change their own behavior. Because they don't change, their situation doesn't change and they get angry. Ergo, we get protests over fast-food wages, student loans and outsourcing, 20-somethings who can't figure out why they can't land a decent-paying job merely by flashing a Masters in Gender Studies, and so on. Failure is not a consequence of one's own decisions but the product of injustice.
So... "Winning is everything" is not what I am suggesting as the proper approach. What I'm suggesting is that failure should not be rewarded with a prize. Losing should hurt. That hurt should not be dampened with a trophy, but harnessed as an opportunity to teach kids to find the personal shortcomings and environmental variables (such as superior competitors) that caused the failure, so they don't experience that hurt in the future. Giving Jonny a trophy after losing a game – thus ensuring he walks away with something tangible – diminishes the need for this process. He's conditioned to play, not to do what is necessary to win (learn from failure and analyze the competitive environment). This puts him at a severe disadvantage as an adult.