Is the action of the son who refused his wife to go out and greet his father and uncle, because he considered that "mixing," while he remained in the kitchen serving the men, considered Islamically correct, especially since it caused grief to his parents?

Your brother should have permitted his wife to welcome his guests while she was covered, as it is permissible for a woman to serve her husband's guests if there is no fear of temptation. It is narrated in Sahih Al-Bukhari from Sahl that Umm Usayd, the wife of Abu Usayd Al-Sa'idi, was the one who served the Prophet, peace and blessings be upon him, and his companions and gave them drinks. Al-Hafiz Ibn Hajar said: In this hadith, there is permissibility for a woman to serve her husband and those whom he invites, provided there is no fear of temptation and modesty is observed. Your brother must honor his father and strive to appease him, and you all should endeavor to reconcile them.