Yes, you could do this, but it's parasitic behavior that depends on your ability to extract value from the labor of other people's kids. If everyone does this, then there are no kids to take over the work to keep things running and money doesn't do anything anymore because the economy stops working.
It did not. I'm just pointing out reality. Kids are expensive, and it's probably easier to save for retirement if you don't have any. Don't get me wrong: I love my kids and wouldn't trade them for anything, and I have no problem working more and harder for them, but we don't live in the age where retirement depends on having kids anymore.
Maybe you meant to propose to change that, so only people with kids have a right to a government pension, which is certainly an interesting idea, but also doubly painful for people who wanted kids but were unable to have any.
Not having kids yourself but still having a retirement of any kind is a privilege that only exists if most people still have kids, so that their kids become workers who will provide the goods and services you need to keep living in retirement.
Government pension doesn't magically continue to work if everyone chooses the individually optimal solution of not having kids and consequently having a much easier time saving more money for retirement. If everyone executed this strategy, all that extra money you saved would become worthless, because there would be no one to give you goods and services in exchange for it.
It was the case in the distant past when your kids were your retirement, and even then people tended to work until they died, but that time is now in the distant past. At least in the industrialised world. There may be poor countries where this is still very much the case.