There is a little trick to bringing up CX award availability.
The reason that the CX availability is not coming up is because there is BA award availabilty on the route you are looking for, and the BA availability effectively "trumps" the CX availability on the booking engine.
By searching for the route JFK - TPE (Taipei), you are effectively looking for the routing New York - Hong Kong - Taipei. As BA doesn't fly this entire route (ie. the HKG - TPE sectors) the booking engine automotically defaults to searching for partner airliine availability, and the CX availability between JFK and HKG will appear.
To actually book the route you are seeking on CX you will of course need to telephone BA.