OK first off, the person who invented that phrase (Carl Sagan) specifically decried the manner in which you are using it, and made that the basis for disowning his own quote.
Second, the logical and reasonable level of proof required in both the enumerated cases should be exactly the same. If it is not, the thinking is simply not based in logic.
A claim that may appear ordinary on its face may turn out to be quote extraordinary upon inspection, and vice versa.
Otherwise a single claim could be both true and false at the same time, depending ONLY on whether it was considered ordinary at the time the claim was made.
Scarborough’s claim is inherently extraordinary until proven otherwise.
All claims are not equal. The function/mechanism of support may be the same, but the quanity/quality of evidence varies based on how uncommon the claim is.
My car is red. Here’s a picture of me with it. Sufficient proof.
Invisible Pink elephants are constantly dancing around us. Here’s a picture, but you can’t see them.