Grasping the practical usage of Central Limit Theorem I've been reviewing the Central Limit Theorem in statistics and I'm having trouble intuitively grasping how it can be practically used. The theorem tells us, subject to constraints, that the distribution of the sample mean will be approximately normal, even if the population distribution is not. From what I understand, this remarkable fact now allows us to use inferential statistical methods that apply to the normal distribution to answer questions about the population distribution. But I struggle with intuitively grasping why this is possible? That is, if I want to answer some question about a random variable X of unknown distribution, how does working with the sample mean of X get me to the answer about X? Is there a step missing that allows me to infer properties of X from properties of the sample mean of X?

You are quite right. In a way, the CLT smoothes out all characteristics of the distribution of $X$, as you always end-up with a normal. All information is lost but $\mu,\sigma$.
On another hand, the CLT is very powerful in studying the properties of the mean of a sample quantitatively, even without knowledge of the distribution. Like Chebyshev's inequality, it is universal.
