This seems’s like an issue created by congress. the constitution only requires a headcount by state. Maybe they should use another mechanism to collect demographic data. Since the concern is not about representation, but allocation, tax returns seem like an obvious alternative and they are already private and collected at a much more granular level.
I don't think the question "Has this person given birth to any children in the past 12 months?" would look good on a tax return.
Isn't that already on the tax return? Your dependent count would increment from the prior year. The IRS can also distinguish births vs adoptions and step children by the checking for novel SSNs.
My home country pays a baby bonus to people and it's administered via the tax system, so I think we ask something very similar actually.
Have you filled out a federal income tax return in the US?
It absolutely asks for the names (and SSN) of any dependents. It's trivial to infer whether one of the adult(s) filing the tax return gave birth in the last 12 months based on the last 2 years of tax returns for those adult(s).