It doesn’t what? Prove that he was born in the US? It says right on it that the place of birth was Honolulu, Oahu County, State of Hawaii.
There is this rule of evidence call the best evidence rule. The best evidence is what is available to prove the truth of whatever one is trying to prove. Copies of documents can be admitted into evidence if that is all there is, or there are so many copies out there the contents are beyond dispute.
In this case, the assertion is birth in HI. If only a copy is available at a reasonable cost and effort, then the court would accept it as evidence.
My 26 years of experience in law says no court would accept this scanned manufactured COLB as evidence because getting the best evidence is nearly effortless and cheap to produce. Therefore, it is no evidence at all to prove that which it asserts to prove.