The banking system in Taiwan seems to be built on a model of having to inconvenience the hell out of the customers in order to achieve a false sense of security and peace of mind...if you needed to do any banking and forgot to bring that little stupid seal, sorry, no dice.
"But I'm me! Here are my ARC, passport, NHI card, and driver's license complete with photos! I can recite my address, date of birth, landline and cell numbers forwards and backwards, in English, Mandarin, Taiwanese, and Hakka! I can sign my name in English, Simplified Chinese and Traditional Chinese with both my left and right hands! Give me my money!"
"Sorry sir, there is nothing we can do for you without the seal. We can, however, file a form for you claiming that you lost the old seal and need a new one to replace the old one, but that would require that you go out and get a new seal and come back in time before the bank closes at 3:30pm..."
Anyways, rant aside. If you are only planning on using your bank money locally, then I would go with Chinatrust. They seem to work with a lot more merchants and you can get discounts when you charge things to your Chinatrust card (I believe Visa debit cards count too). Plus, Chinatrust ATM's are just about everywhere. Although I do believe that their tellers would only process transactions of more than NT$30,000 in person, and for anything less than that you have to use their ATM's.