No, it does not do a cut off. You can set the voltage on that buzzer, if you set it at 3.2v then you still have some juice to drive the car home when the buzzer goes off.

Grapner has something like this:
https://www.ebay.de/itm/Graupner-LiP...-/292211012346
This is a full voltage failsafe module keping the ESC on a lower power when the bottom voltage has reached but for a few bucks more you can buy a modern brushed ESC with a LiPo cut off like the Hobbywing.