From what I've read it will take a lot more than just economic incentives to put a dent in this problem. The hypercompetitive nature of education and your standing in the workplace and society (leading to overworked students&employees, stress, burnouts, suicides, etc.) in South Korean culture seems to be a big part of the problem, but the politicians don't really seem to touch that aspect of it at all. I don't think just throwing money at it will accomplish anything, they need to sort of rework the way they look at what school & work are all about in the first place.. In a way they need to rebuild large parts of their society from scratch, and they don't seem to even want to consider any of this as a solution.
Dropping birthrates seems to be a problem in virtually every developed country. Here in Canada economic incentives aren't enough to push people to have more kids either, even though we lack some of the same problems South Korea has with an overworked and overstressed population and overcompetitive approach to everything. IMO if you want more people to have kids you need to provide them with more money but also more TIME, at the very least. Two parents having to work 40-60 hours a week each leaves little time to raise a family unless you want to go completely insane in the process. I am not surprised people are de-cluttering their lives as a result and making them simpler - and no kids makes things a LOT easier. In places like South Korea people also have less and less time to date in the first place, meet prospective partners, etc.
Overall I think it was a massive mistake to model our entire economic model on the idea that the population & everything else will always grow. You can't have unlimited growth, that's just not possible or even sustainable.
We need to rework how our economy works if we really want to fix any of this, but of course that's never going to happen either.