There's a reason why we don't think strategically, and it's actually a very good reason and is unfortunately why we will never have an innately strategic mentality: cost. Specifically, the cost of time. i.e. it's always cheaper in terms of time to make a correct lucky guess on the first try than to work out a solution properly over a significant length of time.

Imagine there was a such thing as a lucky charm, and by holding it, you were, say, 70% more likely to always get the right answer on your calculus test without even needing to completely understand the problem. In this situation, taking the calculus test would take you just a few minutes, and you'd still score well enough to pass the class. In fact, you could take the entire years worth of tests, perhaps, in the same amount of time that it takes the rest of the students to work their way through the first one, yet still most likely pass. Your lucky charm didn't give you the best grade, but it allowed you to quickly solve all the problems you needed to solve and now you can spend the rest of the year taking other classes.

Well, the thing is, the human mind has evolved just such a "lucky charm", specifically our highly sophisticated pattern matching ability. We can look at situations that we've seen in the past, and can generally make a "mostly right" choice much of the time with very little effort or thought. Those humans who had a particularly powerful patterning matching ability were able to "coast" through even incredibly complex situations with a small effort, leaving them more resources available to survive and propagate while those who spent more time working out far more optimal solutions would find themselves sorely behind in the evolutionary race, even though they are ending up with better answers.

Why is time so valuable? Think about it this way: Imagine you could mathematically work out the winning lottery numbers, but it would take you 50 years, or you could guess every week and never win the jackpot, but, on average, could make a few bucks consistently. Which approach will keep you in food and shelter until you reach the age that you can have children? The jackpot may be orders of magnitude more money, but you need the monetary resources up front in order to pay your way through survival.

Evolution doesn't seek out the optimal long-term solution, it seeks out the "just barely good enough for right now" solution. Unfortunately, as long as we continue to evolve in a world where time is as or more valuable than total available resources, we'll most likely never reach the point where "strategic thinking" is something we do by default.

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