Based on the screenshot of the model that you sent, the use of ‘Gamepad’ blocks for the external mode simulation is causing the problem.
During external mode, you cannot have gamepad joystick or gamepad button blocks connected to your motors and then use the Toggle blocks connected to these gamepad blocks to control the motors. What you must be doing is replacing the gamepad blocks directly with a Constant or Toggle block to provide input to the motor during external mode. You can see in my training video for External mode (link) at around 3:15 the model that I use to demo external mode workflow. Notice how I don’t have gamepad blocks in that model. The reason for this is, as Anusha mentioned, the VEX gamepad is not connected to the robot (over USB cable or wirelessly) during external mode, thus the gamepad blocks in the model during external mode will NOT work. The toggle blocks that you currently have connected to the gamepad button blocks in your model are for NORMAL SIMULATION purposes only. They will not work during external mode to provide input to the servos/motors.
So, an example model for external mode, in your case, would look like the image below. You will have to modify your entire model to follow this design for external mode.
(Notice, how unlike your model below, I don’t have a Gamepad Button block connected to the Switch input port.)
Running this model in external mode should now let you use the Toggle block to change the input to the Servo and thus you should now be able to control the servo angle on the robot.
For downloading to the robot….
Once you are done with the External mode test, you must change your Simulink model to INCLUDE the gamepad button block in the model before deploying the FINAL ready program to the robot. So, when it is downloaded, you should be able to connect the VEX gamepad to the VEX cortex (wirelessly or using USB cable) and control it using the buttons on the gamepad. The model for downloading should look like this:
Try out the suggestions above and let me know what you find. Hopefully this will resolve the confusion that you are having.