Yes, everyone knew that Obama’s father was never a US citizen in 2008 - but that substantive issue was never able to be brought up in federal court, as all of the courts said that anyone who brought up the issue lacked standing to do so. The simple fact is that we’ve NEVER had any real safeguard to make sure that a person running for President or Congress was actually eligible.
That bit about Lincoln backing Fremont is what is known as a “red herring.” Lincoln was a politician, not a Supreme Court Justice ruling on an active case or controversy. His (and his Party’s) narrow self-interest in 1856 do not - CANNOT - override the Constitution.
Undefined terms in any law are defined by the prevailing law of the jurisdiction at the time the law is passed. The prevailing law at the time the Constitution was passed/ratified was the English Common Law, which was the basis of our entire legal system. Literally thousands of Supreme Court cases have confirmed that over the course of the last 200+ years.
The “Common sense” meaning of any words now is not necessarily the same as the common sense meaning of those same words at the time a law (or Constitution) became the law of the land. When in doubt, our judicial system ALWAYS looks to the ORIGINAL meaning of words or phrases.
In short, I think that you are incorrect...which doesn’t mean that I wouldn’t agree with using the “common sense” meaning of “natural born citizen” if we were drafting an amendment to the Constitution. However, we are NOT doing so, and are left to interpret words that are 225 years old AS THEY WERE UNDERSTOOD AT THAT TIME.
The Court has a lot of leeway in how they choose to interpret the Constitution (including the 14th Amendment, newer than 225 years). But, given all that has happened in the 225 years, I don't see them going the Vattel route, even if it's technically correct (I don't think it is). At this point, any Supreme Court decision making Obama ineligible solely on the basis of con-law would be seen as contrived, as judicial fiat, judge-made law. It just wouldn't do, and the justices know it, so they won't do it.