Why don't sociologists use more GMM?
General Method Of Moments vs Ordinary Least Squares

You can't think of these as substitutes. One of the classical assumption for linear regression is E[X'e]=0 which gives you a moment condition from which you can apply GMM. Whether your do GMM or MLE/OLS, your estimator is beta = ((X'X)^1)X'y. GMM is less efficient than the MLE because it makes weaker assumptions (no need for normally distributed errors) but is also more robust in that sense. In large samples, it won't make much of a difference.
GMM arose in economics because it is a natural way to estimate rational choice models. Economic models often have utility/profit maximizers. To maximize mathematically, set the derivative equal to zero  there's your moment condition, and you can apply GMM to estimate the parameters based on the assumption that the moment condition is true. This is not typically how sociologists think about the world, so moment conditions do not generally arise in sociological models.