Right
欠 — a pictograph of a person leaning back with mouth open, indexing
歉 in the open-mouth family:
歌 sing,
欢 cheer,
欲 desire,
吹 blow.
欠 specifically carries the 'lack, owe, be in debt' sense — exactly what an apology acknowledges. When you
道歉, you are admitting you owe someone something, even if only the right words.