The USAF, no it has no say, but the courts? If the no religious test clause has any meaning this should be declared unconstitutional.
I strongly suspect the courts will side with the legal reasoning in the article, and issue a ruling saying the law doesn’t really mean what it says. Courts do that all the time, and it is sometimes a worthwhile thing. After all, the person who updated the AFI probably didn’t have access to legal software to help track down every court case or exception - all they had was the text of the law itself.
Frankly, the ruling ought to say the words are a religious test and are unconstitutional.
My only point, as someone who has sometimes had to draft revisions to a regulation, is that the USAF’s hands are tied. The USAF has no business deciding the words are unconstitutional and must be stripped from the law.
Now, if I were the officer administering the oath, I might not catch an omission of those words by the oath-taker, if you know what I mean...