I agree. It's ridiculous that I can't bring shampoo on an airplane, and grandmothers get searched at the airport, but someone can run for president without providing open access to their birth certificate and education records.
Since determining the citizenship of a candidate for the Presidency is a provision inside the US Constitution, my guess is that the responsibility for ensuring proof of citizenship is an implied responsibility of the Supreme Court. They would then hand enforcement responsibilities of their findings/rulings over to the Justice Department. The openly partisan nature of the presidency and the congress would also argue for this being a Scotus responsibility.