I'm not a Bluetooth expert, but my guess would be that Bluetooth Serial is a BT Classic protocol. The ESP32C3 only supports BTLE.

Wow, this should be the reason. Thank you !!!

I am having trouble using the example of ESP32C3. The used sketch is shown as follow:
#include <BluetoothSerial.h>

#if !defined(CONFIG_BT_ENABLED) || !defined(CONFIG_BLUEDROID_ENABLED)
#error Bluetooth is not enabled! Please run `make menuconfig` to and enable it
#endif

BluetoothSerial SerialBT ...