You can't take the birth certificate out of the discussion. For about 90% of Americans, the birth certificate is the document used to prove citizenship. In some states, the birth place of the parents doesn't even appear on the child's birth certificate. In states that do list the birth place of the parents, they are not required to submit evidence of their place of birth. In other words, everyones birth certificate would be worthless as a proof of citizenship.
You've just managed to prove to yourself why the birth certificate is ALREADY not accepted by itself as proof of anything, despite your naive belief that 90% of Americans have nothing else to work with. Where are you even getting that statistic? From the smell, I'd say out of your ass.
The whole idea of striking section one of the 14th amendment is a bit ridiculous. It is the basis for the Supreme Court ruling that struck down segregation as well as other rulings. A new amendment to replace section one? That would require a 2/3 vote of each house of Congress plus ratification by the state legislatures. It would also require new laws in most of the states related to birth certificates and proof of citizenship. You are more likely to see UFOs on the lawn of the white house than any of this.
First of all, I don't think anyone's talking about "striking down" any of the Fourteenth Amendment, so much as clarifying it to reflect the changing situation in our country, which is what the purpose of the Amendment process IS. In fact, the 14th Amendment itself was an attempt to clarify the Constitution to reflect the changing situation in the country in 1896.
Second of all, the vision of the 14th Amendment as this impenetrable bulwark which forced the end of segregation is sort of funny, given that it was ratified in 1896 and the Supreme Court and lesser courts promptly began handing down decisions for almost a hundred years to the effect that the 14th didn't really mean much of anything. The basis of the Supreme Court ruling against segregation - Brown v Board of Education, in case you didn't know - was much the same as the basis for any Supreme Court ruling: the Supreme Court decided it wanted to rule that way.
I'm rather confused on why you think the Amendment process is so unlikely to happen, given that it has happened 27 other times, if I remember correctly, and for matters a lot less intense than this.