How can I turn back on the lift gate control on a replacement remote?

You might need to go back to the dealer but let's try the procedure below first.

Programming Transmitters
Transmitters can be programming in various ways using the procedures outlined below. Using the Replacing Transmitters procedures will first erase all the known transmitters from the vehicle. Any existing transmitters and any new transmitters will then be programmed. This procedure should be used any time a transmitter is required to be unlearned or erased from a vehicle. If a new transmitter is being learned to a vehicle to replace a damaged, inoperative, or stolen transmitter, the Replacing Transmitters procedure must be used. This ensures that the old transmitter cannot be used to access or start the vehicle after programming.

The Adding Transmitters procedure does not erase any transmitters prior to programming. The procedure will simply program the transmitter into the next available slot. The Adding Transmitters procedure should only be used when adding an additional transmitter to the vehicle. The Adding Keys procedure should never be used to program a transmitter to a vehicle that is having a transmitter replaced, regardless of the cause for the replacement.

Replacing Transmitters (Without SPS)
Note:
This procedure will unlearn all previously learned transmitters. All transmitters that are to be programmed must be with the vehicle at the time of service.
This procedure will take approximately 30 minutes to complete.
A total of eight transmitters can be learned to a single vehicle.
This procedure will only learn the vehicle transmitter information. This procedure will not learn any immobilizer information between the body control module (BCM) and engine control module (ECM).

This procedure may be used with or without existing learned transmitters being present. If existing transmitters are present, keep the transmitters away from the vicinity of the vehicle. Failure to keep learned transmitters away from the vehicle will result in the learn changing to the Adding Transmitters procedure.
Insert the vehicle key into the key cylinder on the driver door handle and turn the key, counterclockwise, to the unlock position five times within ten seconds. The driver information center will display Remote Learn Pending, Please Wait.
After ten minutes, the driver information center will display Press Engine Start Button To Learn. Press the ignition mode switch. The driver information center will again display Remote Learn Pending, Please Wait.
Repeat Step 2 two additional times, for a total of thirty minutes. All previously known transmitters have now been unlearned. Remaining transmitters can be relearned during the next steps. The driver information center display will now display Ready For Remote # 1.
Note: The transmitter pocket can be found in the following locations:
Chevrolet Equinox †front cupholder.
GMC Terrain †driver side cupholder.
Holden Equinox †passenger side cupholder.
Chevrolet Blazer †front cupholder.

Place the new transmitter into the transmitter pocket.
Press and release the ignition mode switch. When the transmitter is learned, the driver information center will indicate that it is ready to program the next transmitter.
Remove the transmitter from the transmitter pocket and press the unlock button. To program additional transmitters, repeat Steps 4 through 6.
When all additional transmitters are programmed, press and hold the ignition mode switch for 5 seconds to exit programming mode.
Replacing Transmitters (With SPS)
Note:

This procedure will unlearn all previously learned transmitters. All transmitters that are to be programmed must be with the vehicle.
This procedure may be used with or without existing learned transmitters being present.

This procedure will take more than 15 minutes to complete.
A total of eight transmitters can be learned to a single vehicle.
This procedure will only learn the vehicle transmitter information. This procedure will not learn any immobilizer information between the body control module (BCM) and engine control module (ECM).

If the battery voltage is low, charge the battery before continuing with the procedure.
Connect a scan tool to the vehicle and access SPS.
Ensure that all power consuming devices are turned OFF on the vehicle.
Select the SPS application and follow the on-screen instructions.

Select Reprogram ECU.
Select IMMO Immobilizer Learn - Setup.
Select the Program Transponder or Remote Key (Delete) function.
Note: At multiple times during the learn procedure, SPS will instruct you to turn the ignition to the run position. Make sure the vehicle is actually in the run mode before continuing on the SPS terminal. If the igniting is not in the run mode, the learn procedure will fail. To verify the vehicle is in Run mode, verify the green LED is illuminated on the ignition mode switch.
Note: The transmitter pocket can be found in the following locations:
Chevrolet Equinox †front cupholder.
GMC Terrain †driver side cupholder.
Holden Equinox †passenger side cupholder.
Chevrolet Blazer †front cupholder.

Follow the on-screen instructions.
After programming all transmitters, Programming Complete, is displayed.
Press and hold the ignition mode switch for 15 seconds.
Press the lock and unlock button on each transmitter that was programmed. This will awaken each transmitter and allow passive and active keyless entry functions to be established.

With a scan tool, clear any DTCs.
Verify each transmitter that was programmed is operating properly. Operate each of the keyless entry functions using the buttons on the transmitter and then start the vehicle. When verifying operation, make sure that no other transmitters are near the vehicle.
Adding Transmitters (Without SPS)

Note:
To initiate, this procedure requires that 2 learned transmitters be available.
A total of eight transmitters can be learned to a single vehicle.
This procedure adds transmitters only. The procedure does not erase previously learned transmitters.

Place both of the recognized transmitters in the vehicle cupholder.
Insert the vehicle key into the key cylinder on the driver door handle and turn the key, counterclockwise, to the unlock position five times within ten seconds. The driver information center will display Ready For Remote #2, 3, 4 or 5.
Note: The transmitter pocket can be found in the following locations:
Chevrolet Equinox †front cupholder.
GMC Terrain †driver side cupholder.
Holden Equinox †passenger side cupholder.
Chevrolet Blazer †front cupholder.

Place the new transmitter into the transmitter pocket.
Press and release the ignition mode switch. When the transmitter is learned, the driver information center will indicate that it is ready to program the next transmitter.
Remove the transmitter from the transmitter pocket and press the transmitter unlock button. To program additional transmitters, repeat Steps 3 through 5.
When all additional transmitters are programmed, press and hold the ignition mode switch for 15 seconds to exit programming mode.
Adding Transmitters (With SPS)
Note:

This procedure may be used with or without existing learned transmitters being present.
This procedure will take more than 15 minutes to complete.
A total of eight transmitters can be learned to a single vehicle.
This procedure will only learn the vehicle transmitter information. This procedure will not learn any immobilizer information between the body control module (BCM) and engine control module (ECM).

If the battery voltage is low, charge the battery before continuing with the procedure.
Connect a scan tool to the vehicle and access SPS.
Ensure that all power consuming devices are turned OFF on the vehicle.
Select the SPS application and follow the on-screen instructions.
Select Reprogram ECU.

Select IMMO Immobilizer Learn - Setup.
Select the Program Transponder or Remote Key (Add) function.
Note: At multiple times during the learn procedure, SPS will instruct you to turn the ignition to the run position. Make sure the vehicle is actually in the run mode before continuing on the SPS terminal. If the igniting is not in the run mode, the learn procedure will fail. To verify the vehicle is in Run mode, verify the green LED is illuminated on the ignition mode switch.
Note: The transmitter pocket can be found in the following locations:
Chevrolet Equinox †front cupholder.
GMC Terrain †driver side cupholder.
Holden Equinox †passenger side cupholder.
Chevrolet Blazer †front cupholder.
Follow the on-screen instructions.

After programming all transmitters, Programming Complete, is displayed.
Press and hold the ignition mode switch for 15 seconds.
Press the lock and unlock button on each transmitter that was programmed. This will awaken each transmitter and allow passive and active keyless entry functions to be established.

With a scan tool, clear any DTCs.
Verify each transmitter that was programmed is operating properly. Operate each of the keyless entry functions using the buttons on the transmitter and then start the vehicle. When verifying operation, make sure that no other transmitters are near the vehicle.

Please let us know how it goes

You are welcome, please let me know how it goes and have a great weekend as well.