I'm "that guy" in the video. That's me. I'm just sitting in a chair where my sensors can see me. All you need is the headset (which you have already, don't you?), and I'm using a controller, which makes flying mostly easy (thumbsticks suck for fine adjustments though). Left stick is ailerons and elevator, right stick is rudder and throttle. Right trigger fires. That's basically all you need to play. The only other button I use is Y (top one, yellow) to switch between cameras.
But yeah, if you don't want to use a controller and don't have a joystick, you're kinda out of luck. A regular joystick will work fine though. You'll never have to take your hand off of it, and you just need to keep your other hand's fingers on WASD to handle the rudder and throttle.