More concretely, let's suppose you're 92 and you need full time nursing care. You still have retirements savings so should be able to pay for it. Now let's say there are more people needing nursing than there are nurses. What happens to the cost of that nursing? It goes up until some cannot afford to pay and demand balances supply.
Without a working age population, your retirement assets will be inflated away into worthlessness.
So far rich world governments and individuals haven't found an optimal incentive/cost share ...
The cost does not change. This situation that you describe is already present a many countries who have an acute nurse shortage. The gap as always is filled by immigrants from the global south.