Whether those public funds and private options will be enough, of course, is a different question.
When you're investing in a private fund, or in real estate, you're still relying on your children. The material wealth you are consuming is still being produced by the generations that come after.
If your children's generation is poorer than your own and if there is less of them, then there is nothing you can do except rely on the children of people in other countries.
The government could just mandate old buildings get destroyed and turned into parks.
Remember, >55% of people are homeowners - they'll vote for anything that will protect the value of their homes - especially if non-homeowners pay for it entirely.
Governments can always find ways to limit the supply of housing. They've been doing it fabulously in most of the West for the last 30 years.
Two reasons why I don't think there will be a supply/demand mismatch:
1) Housing doesn't simply stay there once it has been built. Koreans seem very eager to tear down "old" buildings (anything more than 30 years old), so it's going to be fairly easy to control supply if demand goes down.
2) The number of people per household is steeply declining. Even if the population drops to half of the current level, if there are only half as many people per household, the total number of housing units needed will be largely the same. Smaller units will become more popular, though.