Barry Goldwater was born in Arizona when it wasn't a state. He ran in 1964 and would have won if millions and millions of people had voted differently. Wouldn't it be the same thing?
Well Goldwater probably inherited birthright citizenship from his two American citizen parents, but even if he didn't the real difference there would be that anyone born in the Arizona territory was a U.S. citizen at birth according to federal law. This is true today of Puerto Rico, Guam, etc. American Samoa, however, is an unincorporated territory, so being born there does not automatically confer citizenship.
All US territories with the exception of DC and Palmyra Island are unincorporated. American Samoa's distinction is that it is the only inhabited unorganized territory of the US. However, in the case of the organized territories, that's not the reason the law grants citizenship at birth. Puerto Rico and the Philippines quickly became organized territories after the Spanish American War, but it was only later that Puerto Ricans gained birthright citizenship. The current situation with American Samoa is largely because of what the local leadership desires. If there were any agitation for birthright citizenship or a change from unorganized to organized, I can't see Washington objecting, regardless of which party were in charge at the time.
So why doesn't the local leadership want American Samoans to have birthright citizenship?