Old people, even those who rely on care workers directly, also rely on a lot of other types of workers. They need to eat, so some portion of the farmers, agricultural processors, logistics workers, cooks, dishwashers, etc. will need to continue to support the industries that feed people. Then the industries that feed people also rely on their own supply chains: equipment manufacturers and maintainers, electricity and energy, etc.

Simply being alive relies on the work of others. Broadly speaking, we expect there to be a ratio of workers to the broader population, including those who are not working: children, students, disabled, elderly retirees, etc. If the workers stop working, the non-workers won't be able to live.

If there's a one-person society, they basically will always need to work at least some to stay alive. If they're incapacitated from age or injury, that might mean death, no matter how much they've accumulated up to that point.

So no, I don't think this is a uniquely capitalist problem. Non-capitalist societies have dealt with population collapse before, but those tend to impose real danger to the non-working elderly, and not all of them survive the turmoil.

Population collapse is also occurring in communist countries. China additionally scuppered themselves by previously putting taxes on multi-child families to get ahead of overpopulation and in doing so accelerated the collapse of the working population ahead of that in capitalist countries.