How sound is the story attributed to Caliph Uthman ibn Affan, may Allah be pleased with him, regarding his brother drinking wine and leading the people in prayer, and Uthman's reprimand of the witnesses, followed by the intervention of Aisha and Ali ibn Abi Talib, may Allah be pleased with them both, in the matter?

The mentioned story is fabricated. It has been authenticated in the Sahih collections that Uthman (may Allah be pleased with him) was the one who ordered the lashing of his maternal half-brother, Al-Walid ibn Uqbah. In Sahih Muslim, Al-Walid was brought forward, and two men testified against him that he had drunk alcohol and vomited. Uthman then ordered Ali to lash him. Ali then ordered his son, Al-Hasan, but he refused. So, he ordered Abdullah ibn Ja'far, who lashed him until he reached forty lashes. Uthman said, "The Prophet (peace be upon him) lashed forty, Abu Bakr lashed forty, and Umar lashed eighty. All are Sunnah, and this is more beloved to me." In Sahih Bukhari, Al-Miswar and Abdul-Rahman advised Uthman regarding Al-Walid's matter, and Uthman affirmed that he would uphold justice. Al-Walid was lashed forty times, and Uthman ordered Ali to lash him. Ibn Taymiyyah mentioned that the innovator's saying, "Uthman wanted to suspend the Hadd (punishment) for drinking concerning Al-Walid until the Commander of the Faithful (Ali) enforced it," is a lie, for Uthman was the one who ordered Ali to carry out the Hadd upon him.