I thought he was born in the US, but of a non-citizen father, who wasn't naturalized until well after Chester was born. The allegations of foreign birth never played out. Thus there was no cause for challenging him as not natural, except on the basis of having a non-citizen father. Now of course some have said that would make him not natural born, but there's at least some disagreement on that score and it was not pursued after he became Vice President nor after he succeeded to the President when President James Garfield was assassinated.
But, my "what if" scenario assumed that your "must prove eligibility laws, which I of course support, resulted in revelation that Obama was indeed not eligible. Different situation, different effect.
It was never proven at the time that he did not qualify, but apparently he didn’t qualify (at least, after the fact some say that and say it’s proven). So, it’s possible that Obama may never be proven to not qualify — and maybe “after the fact” (after he’s left office, it may be proven..., who knows...).
The thing that is at issue is that the allegations are not proven (first item) and then (second item) a person takes the office of President (under those continuing allegations of not being qualified [but not proven]).
It sort of sounds like what may be happening with Obama is what happened with President Arthur.
The allegations (for President Arthur) are that he was born in Canada (and now someone else says that there were questions of being born elsewhere, other than Canada, outside of the U.S.). So, there were multiple allegations, apparently.