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.
For example, in 2005 Social Security had a 172B surplus, but the US government had a deficit of 318B. Social Security's 172B surplus was spent by the US government in the same year for government services on the people of 2005. The money was spent on voters, taxpayers, and citizens in 2005, but the liability was recorded against people who hadn't been born yet.
You've referred to a fiction, but reality is as I've stated. The US government loves creating fictions, but that doesn't mean we visit the Vietnam Police Action Memorial.
Raising your child will run you ~300,000 USD, in addition to the massive opportunity cost from time investment. Given that social security is funded entirely through taxes in that tax year, it's a bit much to expect the children of 53% of the population to care for 100% of the elderly when the willfully childless 47% of the population did not put in more upfront in some other way. As birth rates drop, the ratio of workers to retirees will skew worse and worse, and the entire safety net will collapse progressively.
Alternatively, we can encourage and incentivize child rearing. Yes, that means that the willfully childfree will need to pony up resources for those who want children. What would you prefer: pay now for the continuity of the safety net, or the collapse of the safety net?