There are solutions other than pleading with people to have more kids.
It's the US government saying "since we are a wealthy nation, we will not let you spend your elderly years destitute."
That's it. That's literally all it is. It was meant to keep grandpa from being a burden on his family when he was too old to work, which in the 1930s meant when you were too old to do actual physical labor.
Since then the American public has seen it as a nice little bonus they get for living past 65 and has started stealing from younger generations in the form of sovereign debt in order to maintain that instead of treating it like the insurance policy it actually is.
Social Security has it's own funding stream, a tax on wages earned. It does not add to the deficit or increase borrowing. If the funding dries up, the benefits are reduced. This is the Social Security cliff we hear about all the time.
A retirement bonus (which is what OASDI is for millions of retired Americans) costs more than a retirement insurance plan, because one gets drawn from by everyone, while the other can simply be denied if it isn't strictly necessary for the person trying to collect the benefit.
A more expensive program requires more tax revenue to administer, revenue that could be used on anything else. Since we haven't had a real national discussion about federal taxes in 30+ years in the US, we now take the spending that would be funded with the tax revenues that could be freed up by a less-expensive implementation of retirement insurance - one that's treated as actual insurance - and instead fund it with deficit spending.
More complexity is what makes programs more expensive to administer, not size. You are suggesting adding significant complexity (actual insurance) to what is currently a relatively simple program.
There are lots of things that make it more complex. For example, actual insurance requires you to continue paying in each year to get a benefit. Currently, I can take a year off from working and it has minimal impact on my SSI benefits (as SSI just takes and average index of your 35 highest earning years).
The money doesn’t take care of people, people paid by the money do.
I don't know if the same would be true for most people, though (I make significantly more than the average household income -- although probably less than a lot of people here because I'm not making Silicon Valley money -- and I don't have children).
It's an interesting idea worth exploring, but I'm not sure if just switching social security money to retirement accounts would be sufficient.
Administration and making sure people don't dig into them before retirement would potentially be very expensive too (although to be fair, I'm sure Social Security administration costs are very high also).
Are you aware of any studies that analyze such a solution? I imagine there must be something out there.