TARGET installs a virtual device driver into the system. When you activate the profile, it interrupts the normal interface between the stick and system, and redirects the joystick input to a Thrustmaster Warthog Combined virtual stick. You can confirm this by opening joy.cpl (type it into the start menu). Open TARGET and start the script. The T16000M. FCS will disappear and be replaced by the combined stick.
The game client has a Thrustmaster T.16000M default, but it will not load those binds if you run TARGET, as this is basically a different stick the game does not have a default set of binds for. You will need to run TARGET then create a new set of bindings. You can do this from blank, or modify KB/M into a custom set.
If you have mapped the controls with TARGET running, Windows 10 has some rather aggressive power management features. One of these is to cut power to unused USB ports. As the OS can no longer detect the stick plugged into the USB port, it cuts power to it. You can change this with a registry edit. I can explain how to go about this, but it would be a waste of time if it's not actually your problem. If you do continue to use TARGET (it's great for power-pip management and joystick response curves), then you will very likely run into this.