1. As pointed out in the question, there are numerous traditions from the Holy Prophet (S) and his pure Household in the interpretation of the following verse which says:

«إِنَّ اللَّهَ وَ مَلائِكَتَهُ يُصَلُّونَ عَلَى النَّبِيِّ يا أَيُّهَا الَّذينَ آمَنُوا صَلُّوا عَلَيْهِ وَ سَلِّمُوا تَسْليماً»[1]

Surely Allah and His angels bless the Prophet; O you who believe! call for (Divine) blessings on him and salute him with a (becoming) salutation.
These traditions have been reported on the authority of both Shiite and Sunni reporters and it has been stressed in all those traditions that the family of the Prophet (S) should be mentioned while saying “peace and benedictions of Allah be upon the Prophet (S)….”. It has been stated in some of these traditions that if the family of Prophet Muhammad is not mentioned in a greeting, it would be imperfect and deficient.[2]
2.  The Holy Prophet’s method of praying has been reported in Hadith (textual) sources and all Muslims are supposed to adhere to the same method and pray in the same way as the Prophet (S) prayed. Not only in praying but also in all other acts of worship and rules of religion, Muslims should follow the Prophet and should act in the same manner as he acted.
When it comes to prayer, one of the places where[3] greetings are sent out to the Prophet and his family is Tashahhud. As mentioned in the traditions, God, the Glorified, took His Prophet, Muhammad (S) on the journey of Mi’raj (lit. ascension) which was a nocturnal journey. Gabriel recited Adhan and taught the Prophet (S) how to perform prayers in the same way and manner in which we perform our prayers today.  Then Gabriel told him to recite Tashahhud by sending greetings upon himself and his Ahlulbayt (AS).  The Prophet said:

صَلَّى اللَّهُ عَلَيَّ وَ عَلَى أَهْلِ بَيْتِی[4]

Peace and blessing of Allah be upon me and my household.
This clearly shows that the Prophet of Islam (S) used to recite salawat upon himself and his pure household – may peace and benedictions of Allah be upon them.
3. Tashahhud has been reported in the traditions and sayings of the infallible leaders (AS) in different wordings [5] in all of which one has to witness to the servitude and messengership of Prophet Muhammad (S) and also send out greetings upon him and his pure household. The sixth Imam, Imam Ja’far Sadiq (AS) has taught us to recite Tashahhud as such:

«الْحَمْدُ لِلَّهِ، أَشْهَدُ أَنْ لَا إِلَهَ إِلَّا اللَّهُ وَحْدَهُ لَا شَرِيكَ لَهُ وَ أَشْهَدُ أَنَّ مُحَمَّداً عَبْدُهُ وَ رَسُولُهُ اللَّهُمَّ صَلِّ عَلَى مُحَمَّدٍ وَ آلِ مُحَمَّدٍ وَ تَقَبَّلْ شَفَاعَتَهُ فِی أُمَّتِهِ وَ ارْفَعْ دَرَجَتَهُ»[6]

For this reason, Shiite jurists have given verdict as to the necessity of ‘salawat’ in Tashahhud.[7]
Therefore, the Prophet (S) did not innovate the prayers and its parts including Tashahhud by himself. In fact, all parts of the prayers (even the content and how the prayers should be offered) were prescribed by Allah, the Exalted.

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