The built-in barge feature is not ON on the enterprise level, but it is ON on the device level. Built in barge is enabled on 7971 from device configuration menu and the privacy is off.
The same configuration is there on 7961 phones and the barge in feature works fine from those phones.
Verify this configuration:
Choose Service Parameters > Cisco Call Manager Service > Cluster Parameters (Device-Phone), and verify if the Built in Bridge parameter is enabled.
This parameter determines whether the bridge that is built in to Cisco IP Phone models 7940, 7960, and 7970 is enabled. A device with a built in bridge can mix streams, which is required for the Barge feature. The only codec that the Builtin Bridge supports is G.711. Valid values specify On (built in bridge is enabled) or Off (built in bridge is disabled). The system uses this parameter for phones that have default chosen in the Built In Bridge field on the Phone Configuration window in Cisco CallManager Administration. This is a required field. Default: Off.
If you configure encryption for Cisco IP Phone models 7960 and 7940, those encrypted devices cannot accept barge requests when they participate in an encrypted call. When the call is encrypted, the barge attempt fails. Reset or restart the corresponding IP phones in order for the parameter change to take effect.
Choose Device > 7971 IP phone > phone configuration -> Device Information. Built In Bridge needs to be ON and verify if Privacy is off.