" SIN appears to be the only destination that works ex-US, and the only two departure/return airports where it appears to work are IAH and LAX".
No. JFK is fine. If you spend some time there is a lot there. For instance, JFK-DOH-AKL-DOH-JFK (the long, scenic route
) can be had for 2783 RT all as part of the "Business Promo". There appears to be good availability and each way earns ~20,000 QMiles.