Having a large pool of potential workers is meaningless if there is no money to pay them for the work, or to pay for the other necessary elements that go into that kind of care.
The problem is not finding jobs for all the people displaced by automation. The problem is finding paying jobs. In your example, who is paying for all that labor? If it's unpaid, why would anyone do it?
The obvious answer is that it's paid from the pension funds. But said funds are maintained by taxing income. So if your entire labor force is working to support older people by drawing on the pension funds that were originally produced by taxing those same older people, what happens when that labor force becomes old?
Note that it's only a problem so long as you need money to partake in the benefits of automation (i.e. to buy the goods that it produces) - but which is the case for our system today.
Elder care is skilled labor.