I have had no problems at all with using an American credit card in Taiwan.
I have a VISA credit card which was issued by an American bank, and they have been sending the bill to my Taiwan address for the past 7 years! It’s now been two years since 9/11, and the bank that issued my credit card has never told me that there is any problem with living overseas and having an American credit card.
The only problem I had is that by the time the bills came, it was usually already too late to pay them without interest. So a few years ago, I arranged for my credit card to be a “debit card”, which means that the money automatically comes from my American bank account, so now I never have to worry about paying the bills.
Last year, I finally got a credit card issued by a Taiwanese bank, which is a lot more convenient to pay, but the limit is only NT$10,000, which is less than 1/10 of the limit on my American credit card. And I had to have a Taiwanese person be my “guarantor”.