The reason there aren't enough young people is: the "old people" no longer saw a need to have lots of kids.
For most of this country's history, people had lots of kids. They did so in the hope that at least one of them would survive disease and accidents and succeed well enough to take care of their parents when they were too old to work. You still see this behavior in third-world countries.
Thanks to better health care, and income transfer schemes like Social Security, it's no longer necessary for a couple to have lots of kids. So, our fertility rate has actually dropped BELOW replacement. I think it's about 1.9 now, and replacement is 2.1, after accounting for early mortality.
Back in the 80's, the Social Security administration did three projections of economic and demographic trends: pessimistic, intermediate, and optimistic. They used the intermediate projections to determine the tax rate needed to make Social Security solvent for 75 years.
The problem? The PESSIMISTIC projection turned out to be closer to the truth. In some factors, even that was too optimistic. So, almost every year -- the year of reckoning (when the Trust Fund is exhausted) gets closer and closer.
And hardly anyone is paying attention.
One thing that you are not taking into account is the huge sums that are given to non citizens from the social security funds in the form of SSI payments. All a Cuban has to do is set foot on US soil and his income is paid for by your social security money. Haitians and Jews from Russia get paid too and there is potential that people from Syria will be on the dole before too long as well.