> The first is pension funds which are paid into by current workers to pay out for past workers. Pension funds inherently depend on population growth to avoid shortfalls.

This is fundamentally a demographic problem, rather than a problem with the pensions system.

If pension funds worked by people saving their own money for themselves, it wouldn't actually be all that different. At any point in time, there's the productive part of the population, and the unproductive. If the productive part is too small to support the unproductive, there is going to be a worsening of the situation, probably especially so for the unproductive.