> If there's not a supernatural enforcer then there are no consequences to lying/breaking your oath. Interestingly, it looks to me that even atheists are endorsing this false idea here!
Most Christian theology holds that God is the present age is not going smite people for breaking such oaths. I think you are right to frame this ritually not theologically.
I would argue that purpose of swearing on an object is that it demonstrates that the oath is being made seriously, as an oath. Lying and breaking minor promises is socially acceptable in almost all human societies, and in some circumstances it is more socially acceptable to lie than be truthful. An oath is a different sort of object. An oath is something which is not socially acceptable to break, and thus swearing on an object which is deeply important to the oath taker ritually emphasizes this fact both to the oath taker themselves and to society at large.
A modernist view of rituals is that they create Schelling fences[0]. You agree to X, everyone knows you agreed to X, you know that you agreed to X, and you know that everyone knows that you agreed to X. Thus, if you do !X, you know that you have broken an important oath and everyone knows that you have broken an important oath. There is a clear line and there are mutually reenforcing internal and external pressures not to cross that line. Furthermore, if someone tells you to d !X, you have an ironclad reason to tell them no and it is much harder for them to take the rejection personally since it is personal. You can point to a public pre-commited attestation to never do !X.
[0]: https://en.wikipedia.org/wiki/Focal_point_(game_theory)