I'll provide an update on ultimately what happened in my case: I tried checking in online over and over (maybe 10 times), and then exactly 3 hours before the flight, the 'waitlist' magically changed to a boarding pass with a seat assigned. So it was a system glitch. I asked at the check-in desk at JFK why this happened, and while she didn't sound confident, she said "oh, we've heard reports that when people buy third-party tickets (i.e. through a travel agent, not directly from the airline), this sometimes happens."
The airline should try to fix this because this wasted an hour of my time all-told, contacting my travel agent etc. Etihad has some good competitors where I did not experience this unnecessary headache.