I would be grateful if anyone could give me a rough score, and point out some of the more obnoxious mistakes.

After reviewing my own essay, I'd say the main problems are:
- Repeating "competition" and "cooperation" way too often;
- Examples are too short and generalised;
- Doesn't focus enough on examples from industry or government, since those are the more explicitly emphasised fields;
- The points are too discrete, while ideally they must smoothly flow from one another;
- The introduction does not give an overview of the essay (admittedly, it's annoying that you need to mull over the main ideas twice in such a short essay).

The Argument:
The best way for society to prepare its young people for leadership in government, industry, or other fields is by instilling in them a sense of cooperation, not competition.

The Essay:
A society focused on cooperation may appear more humane and benign than one based on competition. Yet this appeal becomes more elusive once we account for the more counter-intuitive boons of a society that stresses healthy competition, a society that does not stigmatise the natural human thirst for supremacy.

Competition does not preclude a measure of cooperation. In fact, it is hard to imagine an individual who can succeed in competitive fields without enlisting assistance from others, without negotiating compromises, or without building sturdy teams. Competition fosters an efficient brand of cooperation that springs out of virile and cunning self-interest. Such cooperation is far less dependent on in-group sympathy, and can help bridge major ideological differences.

Next, the most straightforward path to learning lies through mistakes. Focusing on cooperation to the detriment of competition may lead to overly lax punishment, so young people will not be prepared to face the consequences of lapsed judgement. In effect, societies that adopt the cooperative strategy first may be undermined by less humane societies that will remain competitive. People who expect everyone to be forthright and helpful may prove too easy to exploit for machiavellian psychopaths.

Finally, such a clear-cut dichotomy of cooperation and competition may be needlessly facile. There are shades of cooperation. Cultivating a healthy sense of rivalry is different from producing misanthropic leaders who only wish to stamp their fellow man into the mud. Indeed, competition should provoke a burning desire to improve yourself to defeat your competitors in a fair battle instead of endlessly engaging in zero-sum, beggar-thy-neighbour schemes. Sprinters should outrun their opponents, not hack them to death.

In conclusion, gearing new generations towards competition may be suboptimal for an ideal version of the human race, but this approach remains much more realistic. Competition does not eliminate cooperation altogether, merely emphasising material rather than moral incentives. Overstressing cooperation can lead to an inefficient approach to learning, as it will reduce feedback, glossing over negative consequences. Still, we must strive for this competitiveness to remain fair and reasonable.