Why do the Shi’a associate the progeny of the prophet with him when invoking blessings on him?

 It has been established that the Prophet (S) himself had taught the Muslims how to invoke blessings on him. When the following noble verse,

﴿ إِنَّ اللَّهَ وَمَلَائِكَتَهُ يُصَلُّونَ عَلَى النَّبِيِّ يَا أَيُّهَا الَّذِينَ آمَنُوا صَلُّوا عَلَيْهِ وَسَلِّمُوا تَسْلِيمًا ﴾
God and His angels give blessings to the Prophet. O you who believe, call for blessings on him, and greet him with a prayer of peace.1

was revealed, the Muslims asked the Prophet (S): “How should we invoke blessings (on you)?” The Prophet (S) said:

"لا تصلّوا عليّ الصلوة البتراء."
“Do not invoke ‘incomplete’ blessings on me.”

They again asked: “How should we invoke blessings on you?” He (S) replied: “Say:

اللهم صل علی محمد و آل محمد
Allahumma salli ‘ala Muhammad wa Ali Muhammad.
“O Allah! Bless Muhammad and the progeny of Muhammad.”2

The exceptionally high station of the Prophet’s family {al} (‘a) led Imam ash-Shafi‘i to compose the following famous poem:

يا أهل بيت رسول الله حبّكم فرض من الله في القرأن أنزله كفاكم من عظيم القد أنكم من لم يصل عليكم لا صلوة له."
O members of the Household {Ahl al-Bayt} of the Messenger of Allah! (Our) love for you is an obligation, which God has revealed in the Qur’an.
Your lofty station is such that if one does not invoke blessings on you (while offering prayers) one’s prayer will be of no avail.3

1. Surah al-Ahzab 33:56.
2. Ibn Hajar, As-Sawa‘iq al-Muhriqah, 2nd edition (Cairo: Maktabat al-Qahirah), Book 11, Chapter 1, p. 146 and a similar one in Jalal ad-Din as-Suyuti, Ad-Durr al-Manthur, vol. 5, commentary of Surah al-Ahzab 33:56 has been narrated by hadith scholars {muhaddithun} and compilers of Sahihs and Musnads such as ‘Abd ar-Razzaq, Ibn Abi Shaybah, Ahmad ibn Hanbal, al-Bukhari, Muslim, Abu Dawud, Tirmidhi, Nisa’i, Ibn Majah, Ibn Mardaway, from Ka‘ab ibn ‘Ujrah from the Holy Prophet (S).
3. As-Sawa‘iq al-Mahriqah, Book 11, Chapter 1, p. 148; Shabrawi, Al-Ithaf, p. 29; Hamzawi Maliki, Mashariq al-Anwar, p. 88; Zarqani, Al-Mawahib; Sabban, Al-As‘af, p. 119.