Why wouldn't a self pardon be allowed to stand? There is nothing in the constitution that says a president can't pardon himself?
It is a very obvious oversight and easily leads to abuses of power, but that's what constitutional amendments are for?
There arguably is. The Pardon Clause provides that the President "shall have Power to grant Reprieves and Pardons for Offenses against the United States, except in Cases of impeachment." Here, the inclusion of the word "grant" & the very definition of the word "pardon" become relevant because one cannot really grant something - let alone forgiveness - to oneself. I'd wager that if Scalia were still alive, he'd have a field day with this (&, in that same vain, Gorsuch may very well have one yet).
The obvious question is Why didn't Richard Nixon self-pardon?
Because the DOJ advised him that they believed that he didn't have that authority, & he actually listened.