> Doesn't this presuppose that all the information you need to predict the future of your time series is embedded in the past of those time series?

Yes. But usually this is somewhat valid: There might not be data about the causes in your data, but the model should learn not be be over confident.

> Don't most time series we would be interested in predicting (weather, prices, traffic volumes) tend to respond to things outside the history of the time series in question?

Yes and no.

You really want the forecast to be a probability distribution: 95% of the time it will take you X minutes to get home from work if you leave at 17:30 but 5% of the time there will be disruptions.