"In games that were paradoxically competitive and cooperative at the same time, losers would not be eliminated from the game, but would be invited to learn from the winners."

This is, in practice, what we already have.

Two companies in the same space, A and B, compete. Initially, both grow into their space. Eventually, one does better than the other, and B fails. B dies. What happens? A goes and hires the best parts of B, or acquires them, or many variations on that theme. Most of the remainder go find a job somewhere else, perhaps in an entirely different economic sector. The employees of B are not cast into the outer darkness never to compete again, to starve and die.

Capitalism competes at the level of companies, and companies live in a highly Darwinian live-or-die environment. But the people making up those companies actually live on a different level.

Combine that with ensuring that your business environment makes it easy to start a new company, and the process continues. (Meaning that one of the key things to worry about is ensuring that we do not accidentally destroy that property.)

In fact that's the core brilliance of a limited liability company; all societies and cultures, whether they like it or not, are Darwinian, because we live in a rather harsh Darwinian universe of limited knowledge and resources. Many other social structures have been developed to directly cover it up or deny it, and in the process, fail miserably. It can not be covered up or denied. Since it isn't even caused by a human process, it doesn't care if you vilify it, and it is not amenable to politicking. You can't shame Darwinian competition into not existing any more.

But you can move it. You can move it into entities that are not human. You can move it into things that, when they die, mean a lot of people may feel really bad, but then they by and large go get another job or perhaps retire or worst case live on the largesse of the social safety net that a capitalistic society is wealthy enough to provide, but they don't actually die. You can get the Darwinian benefits of competition increasing fitness, while not actually killing the components of the system. Removing the Darwinian benefits from the system...

... well, actually, it's not like we need to hypothesize about what happens. Have we not already witnessed entities that are so large that they internally function without effective competition and, hypothetically, nothing but "cooperation" inside them, transcending the "system"? Old-school IBM. Many governments. We do not associate these systems with producing paradisaical cornucopias of happiness and resources. We associate them with massive resource wastage, favoritism, law-breaking, and extensive internal infighting. "Cooperation" unconstrained by competition has a concrete historical track record of ceasing to be "cooperation".

If you just "cooperate", you need a mechanism to remove those cooperatives that aren't working properly. People living under the shield of the current system and mistakenly diagnose what the system is actually shielding you from as being caused by the system will always find these methods distasteful. But they are necessary lest the whole system dissolve into useless, wasteful flab. As in biology, so in economic life.