So, my replacement Dual Shock 4 sticks arrived from Reddest China yesterday.
What should have been a simple job turned into a 75 minute odyssey of crawling on my living room floor, looking for a tiny spring, only to give up, sit down, and have it fall out of the controller housing. This was after Round 1 where I had replaced the sticks and put it back together, only to find out the touchpad stopped working. Turns out the touchpad can be opened up, and there's a ribbon cable inside it prone to getting unplugged. Once the dangler was put back in his socket, everything was good to go.
Anyway, that's old and busted at the bottom, new hotness at the top. $2 shipped for 10. Replacements felt slightly lighter but fit perfectly. Ultimately it turns out that Metal Gear Solid: Phantom Pain is ridiculously sensitive to wear in the joysticks, and Metal Gear is impossible if Snake won't stop walking slowly, so that controller has been retired from Snake duty. Works fine with everything else.