After a few days of topping off the rear tire on Mary’s bike, with no gashes or debris in the tire, I finally replaced the Michelin Protek tube and autopsied it:
While it’s possible to extract the valve and perhaps even clean / replace it, I think that’s just delaying the inevitable. The rubber shreds may be necessary to fill large punctures, but they seem to wreck the valve seal.
Her bike now has an ordinary (pronounced “cheap”) tube inside the Schwalbe Marathon Plus armored tire. We’ll see how long this lasts.
The Smell of Molten Projects in the Morning 
An autopsy was performed. What were the results? Did you do the dunk-in-water test? Find a pinhole? Did you carefully examine the tire, looking for a Michelin wire? Inquiring minds want to know!
Outside of the tire eyeball inspection: no embedded objects.
Inside of the tire thumb check: no sharp objects.
Tube eyeball check: no punctures or scuffs.
Conclusion: a rubber shred stuck in the valve!
Although I didn’t like cutting up (what seemed to be) an undamaged tube, it wasn’t going back on our bikes!