If the goal is anomaly detection on time series data, then over-fitting my forecast model on the data is just what I want. If the historical data is fit really well, then I should expect some error from this over-fit model to be large enough to signal an anomaly. Is this true? If not, why so?
No. You can easily overfit without any presence of anomalies in the series. Think of FFT, if you don't smooth it, then the output is a perfectly overfitted series.