Single cell Li-Ion battery - cascaded double protection?
You cannot assume that all built-in protection modules have the same circuit, but you can assume that they will be transparent in normal operation.
The battery should look just like an unprotected battery to your device. Provided the battery has at some time received sufficient charge to 'activate' the PCM, it should stay 'turned on' for discharge. 


If the battery has to be charged to allow discharge then the PCM must have detected a fault condition (over-charge, over-current, or over-discharge). If this happens regularly then you should investigate the cause.


When the PCM is first connected to the battery it may have to be activated. This may be why your circuit is having this issue (each time the battery is removed the BQ2970 thinks it went flat). A battery that is supplied with a built-in PCM may or may not be activated when you get it, depending on how it has been set up and handled.
Devices which use protected batteries that are inserted by the user don't usually have an internal PCM, and those that do have an internal battery which is not user-serviceable. Anyone who plugs a 'raw' battery into a device that is not made for it had better know what they are doing!


However a PCM is only supposed to be the last defense against catastrophic faults. Your device should have its own over-charge, over-current and over-discharge protection to ensure the health of the battery and its own circuits. Your charger should have reliable constant voltage and current limiting, short circuits should not be possible, and the device should shut off well before the battery is completely drained. Do not rely on the PCM to do the job!