I am a dumbass. (Can you say "dumbass" on BikeForums?) But the good news is that I am a dumbass because it turns out that I did *not* snap the brake cable on the Airborne -- I'll take that trade-off any time.

The lighter wheels with the smooth 25c tires are back on the bike, and (without me touching the brake adjustments at all) both wheels are spinning freely when they're supposed to, and stopping when they're supposed to.