The Thief is the noun clause. It is used as the A. SUBJECT of the sentence.
Subject is the who or what the verb is expressing about. The verb confessed refers to the subject, the thief.
It can't be a subject complement because there is no linking verb used.
It can't be the direct object because it is not the recipient of the verb.
It can't be the object of the preposition because there is no preposition written before it.