Many, if not most people who require care in a nursing home for more than a year wind up exhausting their assets. Medicaid then begins paying the home. There is no money left, and any hope that the nursing home resident might have to be able to leave a legacy behind for children is gone.
People do plan for their retirement, of course.
They can also plan for an extended stay in the nursing home. A lot of people, though, can't set aside enough money to do both, but many can shift the risk, not to Medicaid by voluntary impoverishing themselves, but to an insurance company. Long-term care insurance is no longer an unknown or unproven option for paying the nursing home. But many people don't want to pay premiums for years, even decades, and then lose their money if they never enter a nursing home. The insurance industry has created such a product.