The problem is, NOBODY has the job of determining eligibility
Bingo. The 800 pound gorilla in the kitchen.
The Founders intended for the President to be elected in truth rather than form by the Electoral College, which would of course make its members responsible for ensuring eligibility.
Since the EC quickly became only a rubber stamp, the mechanism for determining eligibility defaulted in practice to the voters, who in the last election just weren't interested.
“The problem is, NOBODY has the job of determining eligibility”
But that’s equally true of every other political office. Candidates for US Senate don’t have to produce documentation proving that they’re citizens and at least 30 years old. Candidates for state legislatures, where you often are required to live in your district, aren’t required to produce documentation proving their residence. Affidavits are the accepted documentation for pretty much all offices.
And what’s the check on those candidates? Their opponents, their opponents’ supporters, and the state elections office. If a state legislative candidate suspects his opponent doesn’t meet the residency requirements, he can investigate it and, if merited, file a complaint with the elections office to have him tossed off the ballot. They do take ineligible candidates off the ballot when a legitimate and timely complaint is made. This can even be done on the Presidential level.
The same applies to the President. If there was any legitimate reason to doubt Obama’s eligibility, then a complaint could have been filed during the campaign by his Democratic opponents, or by the McCain campaign, or by anyone in a state that allows individuals to file objections with the Secretary of State. And I’m guessing that the Secretary of States won’t take seriously any argument that revolves around ‘The short-form birth certificate and the long-form birth certificate and the newspaper announcements are all forgeries!’