A friend of mine once brought over a 5200, we played some stuff on it. He had something of like a new in box kinda deal. I know all about the 5200 being junk. We lucky given his deal was like new, the controllers worked!
I actually have a factory case of four new old stock CX-52 Atari 5200 controllers with the latest revision mylar PCBs and they also don't work (only removed to test/verify). That's right: They failed, even sitting on a pallet in the warehouse. The factory tape over the case of four was still intact when I received them.
Turns out that there is more than one way that these controllers typically fail and the only publicly shared fixes out there do not apply to these... though I did find a lot of people encountering this exact failure. The only "fix" I've seen out there is the misguided one from Best Electronics where they would have you miscalibrate the console for other, functional controllers by tweaking an adjustment inside. DO NOT DO THIS! I did figure out a proper fix.
Atari revised the carbon-impregnated rubber membrane buttons and the mylar PCB over and over and over in an attempt to solve the longevity problems with those. That's where they would just stop conducting well enough to register due to oxidation of the contacts or surface carbon wearing off when the rubber membrane deforms under pressure. It's the same kind of construction we use in most controllers today but it was one of the first to use it so it's reasonable that they needed a few revisions to get this right.
By the time they finally solved the issue with buttons losing their responsiveness they had already switched to US-made CTS-brand potentiometers (variable resistors) for the analog joystick (previously used 500k ohm Japanese pots from Alps and Panasonic/Matsushita before). These were clearly existing pot designs that CTS tweaked to meet Atari's specifications since they continue turning well beyond the point where the originals would stop and yet they reach their maximum resistance at the point where the Japanese pots did. That's all well and good as long as they remain 0-500k ohms over that range, but whatever they coated the resistive surfaces with to customize the range becomes more conductive over time. This means that you only get about 325-350k ohms max instead of the specified 500k ohms. That isn't enough for most games to register down or right. With mine you can't move the Missile Command cursor to the right edge or bottom third of the screen. I have a lot of 5200 games and this made every single one unplayable or severely handicapped.
The fix? Well, you can't simply add resistance in-line with the pots or else you will lose range on the top and left sides, but luckily, the console doesn't actually read the resistance of the pot to determine the position. It measures the RC delay of a capacitor. This capacitor charges and discharges repeatedly at a rate controlled by the resistance from the pot, which means we can add a bit of capacitance in-line to tweak the rate. The value needed is probably different for every degraded CTS pot depending on condition and age, but it was a very small capacitor in my case... picofarad or nanofarad range, as I recall. I've been meaning to fix all four of mine on video to help others with this issue but I still haven't got around to it.
There are even more controller issues though. I fixed some CX-52 controllers for a local game shop a month ago and thought I failed since they still didn't work in his console. I brought them home and tested on mone and they worked fine. Ordered a replacement a couple CD4052 analog mux/demux ICs to replace the bad ones in his console and he was good to go.
(my old measurements from early 2017)