What's special about moments that allows "method of moments" to work? The idea behind Method of Moments (MOM) is quite intuitive:
find the parameter values so that the population moments (which are functions of those parameters of interest) matches the sample moments.
But why moments? My hunch is that moments are used because they capture the essence of the DGP. (More formalization of this hunch would be appreciated!) This begs the question: is there "something else" that can capture the essence of the DGP? In other words, can we
find the parameter values so that the population "something else" (which are functions of those parameters of interest) matches the sample "something else"?
Is it correct to think that the likelihood is one such "something else", thus motivating MLE? Or could quantile be one such "something else", thus motivating a "method of quantiles"?

Note that two RVs can have equal $k$-th moments for all $k=1,2,...$ and still be different, so they don't always capture "the essence" of the distribution. There are many counterexamples online but also in Casella and Berger.
However, ff the moment generating function is defined in an interval around $0$, then equality of moments provides equality of distributions.
