A better, cleaner, solution that literally no civilization on earth would ever vote for or want to deal with. "Support families to raise kids" sells way better than "let old people die if they don't have kids to support them."
People are going to die regardless of having supportive kids. The question is who pays for their quality of life while in the final years.
Social Security and Medicare are equally about quality of life and survival. And even if you're okay with impoverished seniors, burdening their children of child-rearing age with a new financial obligation doesn't raise birth rates.
It's better than burdening them with that and FICA taxes and the devaluation of the USD, which are also a financial obligation. The burden can be split amongst children, incentivizing raising more, or parents can opt out of burdening their children by going on a very, very long fishing trip.
The government mandated wealth transfer from young to old is obviously unsustainable, in all countries around the world. It is predicated on the assumption that people will "naturally" opt to raise a minimum of x number of kids (economically productive ones), yet the system is most beneficial to those who raise no kids.
More authoritarian systems have higher variance, even if specific instances might be "better". I use scare quotes around "better" because I would argue giving people democratic power is valuable even if they do dumb shit with it, so you can't just compare democracy to authoritarianism. The latter simply lacks one key thing that democracy gives: some self determination.
This is not a defense of any particular contemporary realization of democracy.
Part of the problem is that the decision to not have children isn't a decision for many people. Some never find a partner (and no, I'm not talking about "incel" nutcases here - I'm talking about countries and regions with a severe oversupply of males), some suffer from medical infertility (e.g. due to injuries, cancer, PCOS, endometriosis), some from genetic infertility (e.g. people with genetic disorders, being somewhere on the wide DSD spectrum or where the partners are not genetically compatible), and some have no other choice than not having children for ethical instead of medical reasons (e.g. both partners are carriers of genetically passed diseases or suffer from mental health issues that make them unable to take care of a child).
You can't just go and punish these people for not having had children in their life, that's just as unethical.