This isn't true, or wasn't true for a long time. SS long had a surplus that it lent to the federal government (a lot of the US debt is debt that is owed to SS). It stopped booking a surplus a few years ago, which means that it is dependent on the government paying back what it is owed, and eventually that will be gone, at which point SS will basically be, as you say, living paycheck to paycheck (or probably be insolvent requiring a cut in benefits or an injection from the general fund).
The rise of AI and automation messes everything up. We don't technically need to have more children, we could even shed a few billion people to a new stable smaller population point. The problem is equitable distributions from AI and automation so that unneeded labor isn't just left to die.