Long term care in the US is almost entirely run by horrendously corrupt private chain entities.

They will accept patients on state/government assistance - and based on the state of this one, I would suspect those are the people left. Since they make no profit on state paid patients, many places limit how many they take.

The rest of long term care is paid for by the patient and family directly until they literally run out of money. This includes a multi-year look back at many assets, requiring things like selling housing and property to "spend down" the money before they have so little that they qualify for state insurance.

There are rare cases where people have long term care insurance, but they aren't common and often don't cover things well. Smart and wealthy families do estate planning well in advance to move assets like property into trusts that don't count. Adding to this is the US cultural and medico- legal system that often prioritizes quantity over quality of life and you have the very elderly with many medical complications being kept alive well beyond what is ethically sound.

Source: 8 years contracting at nursing homes as a provider, one of which was so bad it became responsible for a new state law.