To the evidence you provided that no proof of US birth is needed to run for POTUS.
FYP (I think you intended that to say birth, not just citizenship)
I am confused.
I have yet to see any evidence that says documents demonstrating proof of US birth are required from the FEC.
I understand that it is in the Constitution; however, there does not seem to be any actual mechanism in place that even attempts to verify or validate birth place. This is probably the reason why certain states have enacted legislation now to check birth certificates of Presidential Candidates.