No, the action to exclude the airman from reenlistment barring any other disciplinary or performance issues is illegal. I reenlisted numerous airman during the 80s and 90s and enlisted my son into the army in the 90s. I remember that they were told that when repeating the oath they could omit the reference to God. That may have been unofficial policy at the time but their oaths were considered fully taken. They are swearing an oath to protect and defend the Constitution, not any particular belief.
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.