There’s several different options you can use for an Atari 5200 controller:
The stock controller. It’s non-concentering joystick, mushy fire buttons, and general reliability issues make it kinda suck.
The masterplay clone. There are various adapters for using digital joysticks, like the common Atari 2600 stick, on the 5200. The drawback is that you lose analog capability. For more 5200 games it doesn’t matter, but there’s a handful of games that aren’t fully/optimally playable with a digital joystick.
The PC joystick adapter. Vintage PC joysticks, the kind with the 15-pin connectors, are analog. You can more-or-less use one of these with a custom Y-adapter cable. The advantage is that you get an analog stick that plays all games. The disadvantage is a somewhat ungainly configuration of PC joystick (for movement), 5200 controller (for keypad), and Y-adapter device.
The final option is to build something yourself, to try to do right what Atari did wrong.
All about joystick potentiometers…
A common way of reading the position of a joystick potentiometer is using an RC circuit. If you take a capacitor and charge (or discharge) it through a resistor, then the time to charge (or discharge) a fixed value capacitor will vary with the amount of resistance. It’s simple for a CPU to measure that time and compute the value of the resistor. Substitute a potentiometer for the resistor, and you have yourself a joystick axis.
The value of the capacitor and potentiometer are important to the algorithm. Choose a different capacitor, or a different resistor, and the measurement will no longer correct.
Well, the Atari 5200 stock joystick uses a 500K potentiometer, and the commonly-available thumbsticks of 2018 use 10K pots. That’s a 50x difference. You can’t do a substitution as-is. We’re going to have to play a few tricks to make our 10K pot work.”