When I took my first probability course, it seemed like there were an infinite number of approximation theorems to learn, all mysterious. Looking back, there were probably only two or three, and they don’t need to be mysterious.
For example, under the right circumstances you can approximate a Binomial(n, p) well with a Normal(np, np(1-p)). While the relationship between the parameters in these two distributions is obvious to the initiated, it’s not at all obvious to a beginner. It seems much clearer to say that a Binomial can be approximated by a Normal with the same mean and variance. After all, a distribution that doesn’t get the mean and variance correct doesn’t sound like a very promising approximation.
Taking it a step further, a good teacher could guide a class to discover this approximation themselves. This would take more time than simply stating the result and working an example or two, but the difference in understanding would be immense. And if you’re not going to take the time to aim for understanding, what’s the point in covering approximation theorems at all? They’re not used that often for computation anymore. In my opinion, the only reason to go over them is the insight they provide.