There are a couple of key problems that are apparent in this problem.
1. America doesn't have any kind of organized, non-profit long-term elderly care services available, nor is there a nationwide network of providers and support staff that can be tapped into for struggling areas. These kinds of facilities will only be created when they are sufficiently profitable; considering the location, type of work, and meager pay offered to the most needed of these industry workers, is it a surprise that there is such a shortage? It's not profitable enough to increase benefits and pay to attract talent, so these places lose staff and decline. We need a comprehensive, nationally organized plan to address this problem; both lack of coordination and the profit-motive are hindering this.
2. America doesn't have policies in place to support aspiring families. Long work hours, minimal vacation time, insufficient pay, overpriced housing, expensive healthcare, no paid guaranteed maternal or paternal leave, astronomical childcare costs; with those conditions, how do you really expect people to have more children? If they must struggle to just afford a studio apartment or bedroom in a house with roommates while working over 40 hours per week, then how will they have children and all the expenses associated with them? American policy is extremely anti-natalist.
Bingo. Old people and babies don't have any money and as long as people accept job offers at $45,000 a year instead of holding out for $75,000 a year, they won't ever have enough money to move into a house or at least a two bedroom apartment!