It does not require it per say. You can apply for daughter's US passport first.
But it is a crucial piece of evidence. If I was you, I'd spend 1100 to get the certificate in the next year or two. The reason for that is simple. Sometime in the future, many years later, when renewing passport, your daughter may be asked for some additional proof of US citizenship. If you or your spouse are no longer around (we're all mortals unfortunately), your daughter may have difficulty gathering all your papers and documents to prove her citizebship.
Additionally, this question about citizenship may come up if your daughter ever decides to sponsor a foreign spouse for Green Card.
Those are just two common cases when this may come up. But I'm sure this can also come up if she decides to take up jobs in the US military / government / other agencies.
The best thing is to spend the money, get the cetificate, make copies for yourself (cannot make copies and distribute, it's a crime) and store it in a safe place.