This excerpt is taken from Sheikh Farooq Rafi Sahib's book Qurbani, Aqiqah, and Ashra Zil-Hijjah.
Imam Tahawi, may Allah have mercy on him, claimed that the mentioned narrations, which permit the sacrifice of one goat on behalf of all family members, are either abrogated or specific, and therefore it is not permissible to act upon them. This claim is false and rejected due to lack of evidence.
① Imam Nawawi, may Allah have mercy on him, states:
وزعم الطحاوي، أن هذا الحديث منسوخ أو مخصوص، وغلطه العلماء فى ذلك، فإن النسخ والتخصيص لا يثبتان بمجرد الدعوى
"Tahawi's assumption is that the mentioned hadith (which permits performing one sacrifice on behalf of all family members and including them in the sacrifice) is abrogated or specific, but the scholars have declared Imam Tahawi's claim false because the abrogation or specification of a ruling does not occur by mere claim; rather, solid evidence is necessary for abrogation or specification."
Reference: Sharh al-Nawawi: 122/13
② Abdul Rahman Mubarakpuri, may Allah have mercy on him, the commentator of Tirmidhi, states:
تضحية رسول الله صلى الله عليه وسلم عن أمته وإشراكهم فى أضحية مخصوص به صلى الله عليه وسلم، وأما تضحيته عن نفسه وآله فليس بمخصوص به صلى الله عليه وسلم ولا منسوخا، والدليل على ذلك أن الصحابة رضي الله عنهم كانوا يضحون الشاة الواحدة يذبحها الرجل عنه وعن أهل بيته كما عرفت، ولم يثبت عن أحد من الصحابة التضحية عن الأمة وإشراكهم فى أضحيته البتة، وأما ما ادعاه الطحاوي فليس عليه دليل.
"The Prophet Muhammad, peace be upon him, performing sacrifice on behalf of the Ummah and including the Ummah in the sacrifice is his special prerogative, but the Prophet, peace be upon him, performing sacrifice on his own behalf and on behalf of his family is neither his special prerogative nor is this act abrogated. (The evidence for the permissibility and Sunnah of this act is that) the Companions, may Allah be pleased with them, used to sacrifice one goat on their own behalf and on behalf of their family, as this is a well-known issue. Sacrificing on behalf of the Ummah and including members of the Ummah in the sacrifice is not established from any Companion, and Imam Tahawi's claim of abrogation and specification is baseless."
Reference: Tuhfat al-Ahwadhi: 69/5