That’s an interesting point. It certainly signals that you made a commitment. So in that sense it’s a social promise in addition to potentially being a promise to one’s higher power. It could also be used to signal religious values but that comes down to intent of the individual and seems secondary to the main purpose of making the oath to (ideally) increase integrity.

But I still don’t think making a promise to some set of values that is not believed to have the capacity to enforce them makes much sense. Christians aren’t swearing an oath to Christian values - they are swearing to God. Obviously not everyone is religious, but that’s exactly why it doesn’t make sense to sub in some non religious belief in its place.