I've never heard of that, but it makes sense that they would wear out eventually, though I've not experienced anything but N64 controllers wearing out.
Maybe the N64 controllers weren't so bad after all. They wore out quickly, but at least the movement of the parts inside were directly tied to the movement of the axes, so you would never have phantom input.
Come to think of it, that explanation doesn't explain why the stick positions drift, as in, they move over time but reset to the middle once you move the stick. I still don't quite understand why that occurs.
Actually, most analog sticks used on consoles are custom designed. They all have their own feel, which is unique to that console, and have to be made for certain dimensions to fit within handhelds and similar. Sure, you can get 3rd party ones with a similar feel, but those are clones.
I think you don't hear about it as much on the Xbox One because, in my experience, Microsoft has an excellent returns policy for controllers, at least in the UK. Googling "Xbox One Drift" gives you ample results, that tells me that Microsoft's controllers aren't holy cows, which I already knew anyway from taking them apart. There's a much bigger problem with Xbox One controllers, one that affects most of the faulty ones I've seen, and that's the headphone jack on the new models - it's not soldered to the board, it's a pressure fit sandwiched between two PCB's. Even a mild amount of pressure can dislodge it completely, ruin the pins and make you lose connection, usually permanently until you replace it altogether. I fixed one of them in the past and, try as I may, I couldn't get any contact at all, no matter how I shaped the pins, so in the end I just ripped off the carbon contacts and soldered it directly to the board - fixed instantly. That's a lawsuit waiting to happen because the jack is clearly engineered to fail, it experiences stress each time you take your headset in and out, it should be far more robust.
As to why the sticks center after you move them, it's probably the spring preload moving the wiper to a new spot on the trace that doesn't have a gap in it, or just sudden movement pressing the oxidised wiper harder, causing it to briefly register a new reading. Some people try to clean both with contact cleaner and hope for the best, but the success rate of that is low - it's far less hassle to just install a new stick. Ultimately the way these things work is by utilising the resistance between Point A and Point B - electrons always follow the path of least resistance. If you have gaps in the trace, oxidised or bent contacts or simply dirt in the potentiometer, that path changes, leading to a change in resistance, which in turn gives you an erroneous reading.
Regarding the
"custom" nature of the sticks, nah, most of them are bog standard parts. There's nothing special about them, the manufacturer just picks them based on their properties, like the spring tension, or just the availability. They're not particularly unique, see below:
Xbox One Elite board
DualShock 4 board
Switch Pro Controller board
It's the same sticks, probably from the same factory, they probably sit side by side on the conveyor belt. Occasionally they change the colours on the assembly, I can only assume that the factories colour-code them for some reason, but the side pots are basically the same. In fact, they're interchangeable. They wouldn't go through the trouble of redesigning a part that's already on the market unless they wanted to add features to it - that's expensive!