This article from 1999 I found suggests that South Korea was worried about North Korea's preparedness for Y2K, https://www.deseret.com/1999/12/17/19480898/s-korea-worried-.... That seems to suggest that South Korea itself would have been making sure its own systems were secure.

Is it simply possible that most of their systems were newer than those in other countries, so updates weren't necessary?

There was definitely the possibility of really bad impacts on critical infrastructure. If we had all behaved like Italy then I think it could have been quite bad.

The majority of y2k work I saw on the ground was companies using it as an excuse to upgrade all their kit. I did some assessments and was asked more than once to emphasise the risk a bit more.