Hot line creating circuit with ground
From your description, I'd say one set of old wires in the box was the feed from the power source, and the other set of old wires feeds to other loads downstream.   The newer cable probably feeds the receptacle.  Just a guess though.  


  Update: If I touch the white bundle in this box to the ground, both of the downstream receptacles work, as I expected, but as per this answer, that seems like a dangerous and unsatisfactory solution.  


It's definitely NOT a viable solution - it wasn't really even safe as a test.  However it does indicate the problem is likely due to a neutral connection or break in the wire  between this box and the source.  

Tracing this back and finding the problem may be easy or may be very difficult.