All's well that ends well, I guess. They eventually got me my points back and I just used them to make a brand new booking on an alternate routing, and now everything is showing up correctly on United's end.

So I guess the takeaway here is that even if the airline you booked with gives you an e-ticket number and tells you your ticket is issued and confirmed, it's still possible for the ticket to be invalid and the operating carrier to have no valid travel record for you. TIL.