I feel this thread has some at least partially incorrect info

1. The trains can be very full at many times. Having a non reserved ticket with large luggage simply sucks.
2. If you are willing to pay for the business class the best practice imho is to buy a ticket for a train you feel very safe to make it. For me this is about 2hr after scheduled landing time.
3. If you make it faster to the HSR station you can ask them to change the ticket to an earlier train if available. This way at worst you have to wait some time at the station but you will have a seat and space for luggage.

Personally though I have stopped taking the train back from airport. For me living in Taichung I rather pay a driver 2000nt for door to door service. It ends up being faster even with some traffic. For Tainan though this is not the case.