this is actually an intriguing contest. Think this through to it's natural end.
Federal Government not recognizing State issued Birth Certificates causes a few big problems and many small ones.
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no Social Security Number
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no authorization to work
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no driver's license recognized by States who defend this policy (Red States)
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No flying privilege
So we have Blue States fighting this and Red States agreeing. The non citizens cannot freely travel around the Country by ground air or water. If they are in a job to deliver goods and cannot drive on highways across State lines then the jurisdiction to resolve that problem falls under the Commerce clause of the Constitution.
This argument is lost by Blue States based on that fact.
Example: If a CA driver with a driver's license based on a fake birth cert tries to cross TX and is stopped, the Federal Government has control over that decision