As far as I know, if you want to fly from SJC, you need to take the AA flight to NRT and connect. Otherwise I believe you can get there via LAX, but I do not know if there are other carriers that operate SJC to Asia.

If you fly from SFO, you have more options as I believe AA codeshares on Eva Air direct to Taipei, and on JL or CX with one connection each. For some reason, AA does not fly its own planes internationally from SFO. Of course, UA also offers several options on the SFO-TPE route.