It's YOU who keep doing that, to which I keep pointing out that only a multivariate analysis is appropriate; the authors explicitly named several variables that they included (you can find that list at your link as well as mine).
Your link finds a weaker relationship when newer data is included, and somewhat weaker still when other child tax benefits are included. They conclude that the relationship is "not statistically significant" using a standard approach that treats 'no relationship' as the default (null hypothesis); without that debatable assumption, the actual size of the relationship they find is about a third what the earlier authors did - which would mean that an increase in tax value of child benefits of $440 would (all else held equal) get us at least back to replacement level, and $1800 at least back to our circa-1960 peak.
Such an analysis is only as good as the variables considered.
We have proof that their projections were wildly off so I guess we can conclude they considered the wrong variables.
...the actual size of the relationship they find is about a third what the earlier authors did...
The important part of what they found was the tax incentives affected the timing of births but not the overall number in the medium to long term.
"...which would mean that an increase in tax value of child benefits of $440 would (all else held equal) get us at least back to replacement level, and $1800 at least back to our circa-1960 peak.
Wait. On the one hand you claim a multivariate analysis but the conclusions only hold if "all else held equal"?
Look, whatever they considered their conclusions simply don't hold in the real world.
We've provided much bigger tax incentives than they say are necessary and the rate hasn't improved.