The references I can find on HAC procedures (like Newey-West) in regression focus on the standard error of the estimated regression coefficients and hypothesis testing involving the same. I cannot find any references where the HAC estimate of error covariance is used in a generalized least squares procedure to get 'better' estimates of the regression coefficient. Is this not done? Or are the hypothesis tests just more interesting?
HAC procedures are just about providing consistent estimates of the standard errors. They do not change the estimation of the coefficients. If you have strict exogeneity with serial correlation, your coefficients are unbiased, but the standard errors are incorrect. HAC standard errors address the latter point.
As you allude to, this does not give efficient coefficient estimates. To achieve efficiency, in economics, at least, we typically use a Cochrane-Orcutt/Prais-Whinston procedure. This requires much stronger modeling assumptions to estimate the structure of the serial correlation, however.
They are analogous to Eicker-White heteroskedasticity robust standard errors. This procedure does not alter estimation, it only changes the estimates of the standard errors to ensure that they are consistent in the presence of heteroskedasticity. The efficient fix would be weighted least squares, but this requires modeling the form of the heteroskedasticity.