Is it permissible to return to a wife divorced thrice, with whom a customary (عرفي) marriage contract was previously made, and who has no guardian (ولي), knowing that the marriage was authenticated by the court?

The guardian (wali) is a condition for the validity of marriage according to the majority of scholars. For a woman who has no guardian, the ruler is her guardian. The claim of brotherhood does not make him a guardian. Therefore, this marriage is invalid. The daughter resulting from this marriage is attributed to the husband because it is a marriage of شبهة (doubt/semblance of validity). The divorce resulting from this invalid marriage is effective. A triple divorce pronounced in one sitting counts as three according to the majority of scholars, and as one according to Ibn Taymiyyah.

If you wish to marry her again, there is no harm. It is permissible for a weak woman to appoint a man to arrange her marriage, according to Malik. You both must treat each other well and cooperate to ensure the stability of the family.