There is a simple and well-known reason why successive South Korean governments have not explicitly made such a public commitment in the past: much of Seoul lies within range of thousands of North Korean cannons and battlefield missiles. That vulnerability has provided the North with "escalation dominance," a fact which hasn't changed even if resolve in South Korea may now be stiffening.
As long as Seoul's vulnerability persists, many will doubt the sincerity of General Kim's retaliatory threat. The most important of these doubters would be decision-makers inside North Korea, who seems to relish an exciting game of chicken.
The South Korean government could remove doubts about the credibility of its policy if it got serious about civil defense preparation in Seoul and elsewhere. According to South Korea's Yonhap News, current civil defense preparation consists of a short perfunctory drill eight times per year, plus four hours of video-watching by a civil defense cadre. That's no preparation for a huge and complicated city like Seoul.
Serious civil defense preparation would be highly disruptive, economically costly, and a bit frightening for many. But it would also be a test of how seriously South Korea's population views its problem with the North. A serious public response might do more than anything tried thus far to persuade the North's leaders that future provocations won't pay. And until the South can show some seriousness about civil defense, its promise to retaliate after the next attack will lack credibility.