If it doesn't matter then why did the Democrats use it to justify the adoption of slavery? It's right there in the second link. You probably haven't read it because you choose to remain ignorant.
If it doesn't matter why was it the most requested book at the library in Philadelphia during the constitutional conventions? It certainly seemed to matter to them.
Why did Ben Franklin order another copy from overseas if it didn't matter to him?
"I am much obliged by the kind present you have made us of your edition of Vattel. It came to us in good season, when the circumstances of a rising state make it necessary frequently to consult the law of nations. Accordingly, that copy which I kept, (after depositing one in our own public library here, and sending the other to the college of Massachusetts Bay, as you directed has been continually in the hands of the members of our congress, now sitting, who are much pleased with your notes and preface, and have entertained a high and just esteem for their author."
Source
It certainly seems like it mattered to the framers of the US constitution. It mattered a hell of a lot to them.
Now if the term of art "natural born citizen" did not come from Law of nations, where did it come from?
I challenge you name another treatise on law from the enlightenment era that uses that term.
Face it, Obama usurped the presidency. People who choose to remain ignorant of the US Constitution and it's origins voted him into office.