Why don't I get 5V output on the 5V pin when running off battery?

The 5V pin can only produce 5V if there is a 5V source connected to the board via the USB connector. There is no boost/step-up converter on these boards to provide 5V from the VBAT input.

If you need 5V while operating on battery power for external modules or sensors, you must add your own external boost converter. A good example is the 5V Step-Up/Step-Down Voltage Regulator S7V7F5 from Pololu.