Actually, the later 5200 stock controllers had a rubber boot around the stick that helped with centering. Still not perfect, but lots better than the earlier ones with no boot.
As for calibrating the 5200 controller range of motion, I start always with Missile Command and the controller with the top casing half removed. Basically, I am looking into the guts of the controller and can see the two directional pots. Since I have the top controller housing removed, I have to use a paper clip or similar item to arc the start button location on the flex circuit. Once the game starts, I use the two directional arms to move the Missile Command on-screen cursor back and forth. If, for instance, the horizontal movement is limited to the right side of my TV screen, then I force the plastic pot arm right by a little bit and try the range of horizontal movement again. As you can see, it is trial and error until I get the full left to right horizontal range of motion with the lower (closest to the player) controller pot. Then, I basically duplicate the process with the top controller pot and the vertical range of motion. Ideally, the top (vertical movement) pot should have the pot armature located at 9 o'clock with the on-screen Missile Command cursor visually centered. Also, again ideally, the lower (horizontal movement) pot armature should be located at 6 o'clock with the Missile Command on-screen cursor centered visually. Sometimes, this is all the calibration necessary. However, I always follow by testing the controller next with Breakout. If the Breakout cursor will not move att the to the right or left sides of the screen, then it is necessary to nudge the lower controller pot armature to either side to make that possible. Finally, I test the controller with Gyruss. The trick with Gyruss is that you need the calibrated controller reassembled and the Gyruss ship need to move fluidly through 360° of motion with no stops or hitches. If that does not happen (and the controller still works great with Missile Command and Breakout), then it is time to take the 5200 console case apart and make slight adjustment to the directional movement pot. After adjusting to get the full fluid range of motion with Gyruss, make the controller still operates Missile Command and Breakout with no hitches. Last one to check is the quirky Popeye. Need to make sure Popeye can run the full screen from left to right, and can run up and down all the staircases. if not, the it is necessary to tweak the console movement pot, and even possibly the controller pots one last time.
Lots of wordiness, but I hope this description helps.