The answers to this question on forecast assessment suggest that if the sequence of residuals from the forecast are not properly independent, then the model is missing something and further changes should be made to remove the correlation.
That does make sense to me and it suggests that we should be able to do better than a simple GARCH(1,1) model.
However, in almost all the literature on the subject, this issue is never discussed, and the fact that forecasts produced residuals that are serially correlated is taken as a fact of life. Indeed, people have produced methods for accounting for both serial and contemporaneous correlations when comparing different forecast models.
So, why is this the case? If the GARCH(1,1) model does have such problems, why is it still considered a standard approach for modeling volatility?