Predicting with multiple time series Say you had a set of users, tens of thousands.  You have time series of each of their behaviors using this app.  How might you use these time series to predict future behavior of new users?

The intuitive solution is to feature engineer behavior of users and use that to train a model, such as average weekly minutes in app and things like that.  My issue with this approach is that you lose a great deal of information.

What I'm wondering is if there's a better technique for aggregating a large set of time series data to build a predictive model.  Perhaps an LSTM would work but that seems like it wouldn't capture the nuances of the dataset and I don't believe they're typically used to aggregate predictions on a variety of individual samples.
I would recommend looking into FBProphet.
It's a good starting point for automating the creation of forecasts. It's very easy to use, and often produces better results than classical forecasting methods (ARIMA, Holt-Winters, etc.) right out of the box.
The default settings offer an additive or multiplicative model, comprised of trend and seasonality. This can be a combination of daily, weekly and yearly seasonality patterns, with the option to add custom patterns if required. Holiday effects can also be tracked using the built in holiday calendars, which again can be customised.
This is generally used for data with no external regressors, just a datetime and a measurement. However, external regressors can be configured, to influence the outcome of the forecast.