In all fairness, I have never been a fan of “swearing to God” for secular purposes. From a strict religious point of view, it is “using the name of God in vanity.”
From the start, the US rejected the idea that the constitutional order, leaders and the law were respectively “written by heaven, chosen by heaven, and written by heaven”, so if you disagreed with them you were being blasphemous. This had long been the view in Europe to maintain the noble classes, that they should rule because God wanted them to. God hates peasants.
So even the constitution was clear that it, and the law, were made by men and for men. So could be changed by men without inherently offending heaven.
What is the Air Force Oath? Personnel swearing to support and defend the constitution, not the president, and not their superiors if they give an unconstitutional or illegal order. Bringing God into the equation is just an affront to God.
Typically, swearing before God is only properly done at the events of religious sacrament, in the context of faith.
Even in court, where witnesses used to swear on the Bible, what exactly did that say? “I swear to God that I will follow the secular law written by drunken and corrupt legislators and subject to change”? It is a non sequitur.
God, and the allegiance to God, are supposed to be above the day to day affairs of grubby little mortals.
Secular law was based on Biblical principles. Since the 9th Commandment forbids lying, swearing on the Bible seems an appropriate reminder to the witness that God will be listening to what they are about to say. Naturally this is viewed as an empty gesture in a post Judeo-Christian culture.