Which I mistakenly assumed had squeezed an actual fact into their birther tirade. That site was trying to blame Congress for failing in it's "customary duties". It turns out however, after further research on numerous other sites, that while the Constitution is clear on various factors of eligibility for Presidency, it is rather unclear on who those proofs of eligibility should be presented to, or what action, if anything, should be taken or by whom if a President were found to violate an eligibility rule.

Some of the sites I found angrily indicated that it is the responsibility of the various Secretarys of States to each vet the candidates on their State's ballots but that almost none of them bother and just assume each candidate is eligible if they progress far enough in the nomination process.

I thought Justice Benjamin R. Curtis' comments were interesting (although they pre-date the 14th Amendment):
"The Constitution having recognized that persons born within the several States are citizens of the United States, one of four things must be true:
First. That the constitution itself has described what native-born persons shall or shall not be citizens of such State, and thereby be citizens of the United States; or,
Second:. That it has empowered Congress to do so; or,
Third. That all free persons, born within the several States, are citizens of the United States; or,
Fourth. That it is left to each State to determine what free persons, born within its limits, shall be citizens of such State, and thereby be citizens of the United States."

Apparently, there is a statutory method wherein a President-elect's eligibility may be challenged by Congress, but I found nothing about what to do for sitting Presidents.

It's all very interesting, with very few definitive answers, and I admit it was my mistake in originally only checking a single website about "proof of Presidential eligibility" before posting. It did sound like something that would be probable... like Grant being buried in Grant's tomb.