Because it isn't an AV8R. Seriously man, the moment I took it out of the package I cringed. It was only my inexperience with joysticks that let me even try it for so long.
Yep, the AV8R is certainly one of the more disappointing things Saitek ever made.
Depending on exactly how large the deadzone is, there are things you can do to make it a bit better, though. I myself have removed the centering springs from my old Saitek Cyborg Evo as well as my current Saitek X52, both the stick and rudder twist centering springs. This made both sticks much easier to control around the deadzone, since it removes the mechanical "clankyness" from the (IMHO) poorly implemented centering mechanism.
Of course, modifications like this need to be done carefully so as to not cut any wires or otherwise damage the stick, but this did improve the controllability for me, for both sticks.
As to how not suck with joystick - set deadzone to zero in the game, and sensitivity to maximum. Then, try flying for a while, get a feel to how the ship responds, and if you can't control it, reduce the sensitivity until you find a balanced setting.
The mouse is a different animal altogether. It allows unparalleled precision, but like you have noticed, continuous control inputs don't quite work out.
The key to using the mouse in optimal fashion is to only move it as fast as the ship wants to turn. There is a small delay to the ship's movement as you start turning it, which means if you just yank the mouse to some direction you'll lose some of your precious table room for nothing. And similarly, when you have the ship turning, it only turns so fast. Moving the mouse faster across the surface will not make the ship turn any faster.
You should thus play around a bit, and find out the optimal speed you should move the mouse at. For reference, I can easily turn most ships at least 360 degrees to any direction without lifting the mouse.
You can also combine keyboard and mouse so that you apply continuous control input with keyboard, while controlling the fine precision aiming with the mouse.
Also, the thing that will help you most is practice. Keep at it, Ensign!