According to public law, the words are required. Congress specified the exact wording and gave no one authority to delete it.
I think it is wrong, but this is a problem with Congress, not the military. When the AFI was revised, they simply corrected it to align the AFI with the law Congress passed. The USAF has no say in the matter.
The USAF, no it has no say, but the courts? If the no religious test clause has any meaning this should be declared unconstitutional.
Read the full article at the link. The statute contains a cross-reference to a different statute which says that an "affirmation" can always be substituted for an "oath."