What is the authenticity and ruling regarding the ritual bath (ghusl) of the Mother of Believers, Aisha — may Allah be pleased with her — in front of her brother, for the purpose of teaching them how the Prophet — peace and blessings be upon him — performed ghusl?

The hadith you mentioned was narrated by Al-Bukhari and Muslim. Lady Aisha (may Allah be pleased with her) did not bathe in a way that her entire body was visible to them, for there was a screen between her and them. Rather, they only saw her washing her hair and head, and this is permissible because they are her mahram. One of them is her brother through suckling, and the other is her sister's son through suckling, and there is no harm for a mahram to see that. Aisha (may Allah be pleased with her) intended a practical lesson by this action, as it leaves a greater impact on the soul. One must avoid doubts and their sources, out of caution against being affected by them, for avoiding the causes and apparent manifestations of evil is among the best means of getting rid of it.