Keep after United - escalate to a supervisor, try their Twitter to get attention, whatever works. Just politely but firmly reiterate that it's a UA ticket for travel TPE-JFK. As stated above, it's UA's responsibility to get you TPE-JFK. It may be a different routing or time, but that's irrelevant to their responsibility. For example there is a UA flight TPE-SFO, then they could route you to NYC from there. (No idea what's available, just an illustration.) If you want you could research some available Star Alliance routings before calling to spoon-feed a lazy agent some workable solutions.