Owning a mistake has a counterfeit version: feeling terrible about it, at length. It resembles accountability, costs a great deal, and changes nothing for the person affected.
Guilt is oddly comfortable — it keeps the attention on you. Repair points outward and is far less pleasant: saying the plain sentence, fixing the thing, absorbing whatever comes back.
One is a mood. The other is an action.
If something is nagging at you, ask whether you've actually repaired it or only suffered about it. Then do the smaller, more embarrassing thing and say it directly to whoever it landed on.