Your logic is flawless, not every statement with a sample size of zero is false just because no such event has happened before. However, there are a number of causal relationships being made in the original statement with no reasoning offered and no hint why those causal relationships should hold; as such, it amounts to an unfounded assertion. What does seem to hold is that people see less value in having more children when affluence, welfare and education are in reach. But we have yet to see people stopping to get children altogether because of this.
And again, I think that calling South Korea a 'collapsing' society is quite dramatic. According to the projection offered by https://worldpopulationreview.com/countries/south-korea-popu... they will take to the year 2100 or so (~80 years from now) to undo the population growth sustained since 1960 (~60 years before now), going back from ~52m to ~25m (observe that the curve shown at the address is not zero-grounded so somewhat misleading at first sight); this means going from a bit over 500 people per km² today (rank 26 of 234 regions, so rather high) to 250 people per km² in 2100 (rank 50 in today's terms), roughly equivalent to the population densities of Luxemburg, between the UK (278p/km²) and Germany (233p/km²).
So instead of talking about a 'collapsing South Korean population' without coming forth with any figures we should rather take note that South Korea is thought to have settled on a demographic course that will bring them to a state resembling today's UK, Luxemburg or Germany, within a time span of three to four generations. Whether that will indeed happen is, of course, like any prediction about the future, in principle open; fertility rates could go up or down in the future, and, yes, logically speaking, in theory some day in the future could see the last elderly Korean passing away because no-one cared to reproduce anymore; the Korean Dolly Pentreath so to speak. That's one possibility, however unlikely.
However, for those who shudder in the light of such bleak predictions, let me offer the following observations:
* As noted, South Korea had ~25m inhabitants in 1960. Theirs was a time of population growth, so a time with plenty of young labor force entering the market who could on a yearly basis get educated to enter the many budding businesses and technologies that are responsible for the Korean Wirtschaftswunder. True, but even so, if those 25m had been growing slower or had just stayed at that number, one would assume that South Korea's rise might have slowed to a less meteoric rise, given its unprecedented clip (caveat: 'unprecedented' is own assumption).
* Now if South Korea should maintain its predicted course for some decades, it will be up to the Koreans at that time in the future to decide on their fertility numbers. With the country having to feed and shelter that many less people—be it in the hundreds of thousands or millions—there will be space for more people if need be or the wish arises. Conversely, since unbounded demographic growth is a dead end (because at some point the country will have no more space or offer tolerable living conditions), such growth will simply force the only humane option onto the people of the future, to have fewer or no children, and do it fast.
Seeing as one of the more densely populated places on Earth is embarking on a journey to a less-densely populated future should be seen as a sign of hope, not despair.