I have found Holt-Winters seasonal method a very decent method for forecast, specifically for cases where more recent observations are more representative of the near future. The method equally sounds very promising for imputation of missing intervals. However, I have not found even a single implementation of this method for imputation. Why is that?
My subjective impression is that forecasters and data imputers are simply different people. The data imputers don't know much about forecasting, so they don't think about methods like Holt-Winters. The forecasters, from what I have seen, usually expect their time series not to have missing values. (Yes, that's unrealistic, but as far as I see, they will usually do some ad hoc way of imputation, if they don't throw out series with missing values altogether. They are simply not all that interested in imputation.)
Plus, as Rob writes, to use Holt-Winters to impute values, you'd have to fit the model to the data up to the first missing value, then forecast into the missing period. The problem is that if the first missing value occurs near the beginning of your time series, the model used for imputation will be based on very few observations, so it will be unstable and won't be able to fit seasonality etc. You would not be using the data after the missing period.
Of course, in such a case you could reverse the time series and work backward in time. But that won't help you if you have multiple missing values, some near the beginning and some near the end of the time series.