Korea does have cashless payments with Kakao Pay, SSG Pay and many other Korean "pay" apps...but none that you can use as a tourist. There is no Apple Pay here. There IS Samsung Pay.
T-Money MUST be set up with cash. You are correct - there are many options for residents with bank accounts and postpaid cards. For instance, my check card has the postpaid feature and it operates actually as part of my Korean credit - just pay the balance for, say, August in September.