I was getting black flecks, flakes, particles, in my glass, they look exactly like this guy's
I didn't find a definitely answer online but I did discover the reason I was getting them (so I post this for anyone else who is looking for an answer). I feel like an idiot now, but I had no idea the picnic tap could be completely disassembled. I was opening it up, rinsing, soaking in PWB, soaking in Star-San, and it all "looked" clean. But I finally discovered that the "cap" on the top of the picnic tap will unscrew, and you'll have the base (the path that the beer flows through) in one hand, and the stopper and lever in the other.
Immediately I could see that the stopper was caked in the black flakes that were showing up in my beer. It looks like just dried up old beer gunk. It flakes off exactly like the particles that were ending up in my glass.
Further exploration discovered that with a little upward pressure on the base of the stopper, you can slide the top lever forward, which detaches it from the plunger/stopper and completely disassembles the whole top of the tap. Then you are left with the plunger/stopper itself (an elastic rubbery material where almost all of my gunk was), a spring, the top lever, and the round cap.
Completely disassembled and easy to reassemble. If I had known this, I would have disassembled every time I kegged. As cheap as picnic taps are, it would not be unreasonable to just buy a new one, but my home brew store is an hour away and not exactly convenient. So this is a good solution for me until I can get to the store on other business.
I hope this helps someone else.