One view about retirement in America says that population aging means we need to cut Social Security benefits. One view about retirement in America says that defined benefit pension programs are systematically underfunded and doomed. One view about retirement in America says the 401(k) experiment has failed and individuals don't have nearly enough money. Roughly speaking, I would say that these things are all true. And in fact, while there are pros and cons to all three ways of sustaining elderly people in their retired years, they're also facing the exact same underlying crisis: demographic shifts doom all retirement schemes.
We know the basic idea of savings from the fable of the grasshopper and the ant. The virtuous ant spends all summer stocking food away so he can eat in the winter. And good for him.
The problem a retired person faces is that while you can stockpile food, you can't stockpile human labor. And yet every modern person's consumption bundle consists primarily of services, and that's especially true for elderly people who need a lot of health care services and other assistance. So you can't just stockpile stuff for yourself, you need to stockpile a bunch of extra stuff that you can trade for labor in the future. For example, you might work and save and buy two houses. Then, when you're old, you can live in one house and let someone else live in the second house in exchange for taking care of you. That strategy works great as long as the population is rising rapidly, in which case the ratio of able-bodied workers to houses will be high so you'll be able to trade your spare house for lots of labor. But if the population is stagnant, your house isn't so valuable. If the population is actually shrinking, then one house is only going to be tradeable for a little labor. You, personally, can ameliorate this by saving enough to have two or three spare houses, but if society attempted to do that systematically the problem would only get worse.
In the real world, of course, people save through more sophisticated instruments. But the same problem arises. Stockpiling financial assets only helps if future people are going to want to buy a lot of financial assets. If the rate of population growth slows, your private savings plan faces the exact same problem as a pay-as-you-go public pension scheme like Social Security. Fewer people around to pay payroll taxes means fewer people around to sell your financial assets to. You're doomed.
What if people start having more children? It's too late for that. Even if it happens, it'll take decades for those babies to get through school and start earning enough money to want to buy financial assets. The only viable option, then, is to import adult human beings—immigrants—who'll want houses to live in, who will expand the labor supply, and who will expand the demand for capital goods. This solution won't work forever, since the trend toward population aging is global, but in the short term at least there are plenty of able-bodied adults who are eager to move here and work. Some of those able-bodied adults even have specialized skills in health care professions, and can directly increase the supply of the kind of labor that an aging America will need.