So, Brian basically nailed it, though they are different sized (7/16 tube diameter).
Mine were simply so crusted with mineral deposits the interior valve part was stuck open, thus the water would just stream out. By knowing what to look for, I took them off the tank, cleaned them up and got them functioning. Still a bit of cleanup and perhaps new orings would be nice for general peace of mind regarding leakage.
Turns out the sliding clip had to have been removed intentionally - you absolutely can't reach them, and the airplane-side fitting still has the slot for the clip to engage. But without the clip, if you just plug the tank in until it seats securely on the fittings, that actuates the valve and water can get through to the sink.
Deciding whether I will be worried about leaks over the long term.
