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"?

The idea of the "method of moments" is indeed not restricted to moments, but can be applied to any estimator for a summary statistic. For instance, Elo derived an estimator for chess ratings by approximately solving the relationship between ratings (= model parameters) and mean wins in a Thurstonian paired comparison model (unfortunately, in his book, Elo did not make the relationship clear to previous work by others).
Which summary statistics to use typically depends on what is observable. But even in the original form of the method of moments, there is some arbitrariness: you always have only few parameters, but an infinite set of moments, so there are an infinite number of ways to compute parameter estimators with the method of moments.
