>The problem is the increasingly long span of life when people are not working but still consuming, possibly requiring expensive personal and medical care. The goods and services consumed have to come from someone working somewhere.
This seems like a Lump of Labor Fallacy. Productivity-per-worker-hour has been growing for decades, while we've kept work-weeks and retirement ages the same (or even, in some countries, increased the retirement age). If people work the same amount and productivity keeps growing, there ought to be more goods and services to distribute to the retired as they live longer. And indeed, the per-capita GDP has been growing.
Of course, the generation ratio (retirees to workers) is rapidly becoming problematic, but given the sky-high youth unemployment rates, I don't really trust that's driving the supposed inability to fund pensions either.