This is unfortunate. My initial research shows that a Taiwanese national should not need a transit visa if in China for under 24 hours. In any case, this is something you should have checked with the Chinese consulate and not only the airline. I would also have gotten something in writing about it.
In any case, now you are at the mercy of Delta and China Southern. I am sure they will each blame each other. With Delta's 72-hour cancellation rule, good luck in trying to get your miles back or rebooking.
I would have advised checking TIMATIC ahead of time, not the Chinese consulate. Chinese embassy and consulate websites are years out of date regarding transit visa rules. The airline agents will use TIMATIC to determine whether one can board or not.