[Why the scholars of the school did not mention sending salām upon the Messenger of Allah within the prayer]

Answer (and Allah grants success): What appears to me—Allah knows best—is that salām has contexts and settings in which it is said: he is greeted with salām in his gatherings, in his house, and whenever Muslims encounter him; likewise he is greeted at his grave, and salām is sent to him in life and after death. These are the settings in which salām upon him is especially affirmed.
Prayer (ṣalāh) is not among the settings in which salām is especially affirmed; thus, the prayer includes ṣalāh upon him, for that is emphasized for those present and absent. Salām is a greeting, and greetings are not for the absent and those in his ruling.
Yes, the scholars of the school did mention sending salām upon the Prophet—may Allah bless him and his family and grant them peace—and that is in the wording of the tashahhud narrated by al-Hādī (peace be upon him) namely:
“All salutations, prayers, and pure words are for Allah. Peace be upon you, O Prophet, and the mercy of Allah and His blessings. Peace be upon us and upon the righteous servants of Allah. I bear witness that there is no god but Allah, alone with no partner; and I bear witness that Muhammad—may Allah bless him and his family and grant them peace—is His servant and His Messenger.” … etc.
It may be said that mentioning ṣalāh (sending blessings) can suffice in place of salām, and vice versa. It is well known from the Companions that when they approached the Prophet—may Allah bless him and his family and grant them peace—they would say, “Peace be upon you, O Messenger of Allah,” and in other contexts they would say, “The matter is such-and-such, O Messenger of Allah—may Allah send blessings upon you,” sufficing with one of the two.
Likewise, in the ḥadīth of Jibrīl: “Whoever I am mentioned before him and he does not send ṣalāh upon you …”—and salām is not mentioned there.
What we have said is supported by the fact that the Commander of the Faithful, ʿAlī—peace be upon him—when the Prophet—may Allah bless him and his family and grant them peace—was mentioned in his sermons, would say: “May Allah send blessings upon him and his family.” The same is found in the Ṣaḥīfah of Zayn al-ʿĀbidīn, and likewise in the speech of the earlier Imams—such as al-Hādī, his grandfather, and those before them—who would often suffice with one of the two for the other; this was most common.
Moreover, what is transmitted from the Prophet—may Allah bless him and his family and grant them peace—regarding how to send ṣalāh upon him in the prayer is what the scholars of the school have cited, and it contains no salām: “O Allah, send blessings upon Muhammad and upon the family of Muhammad, and bless Muhammad and the family of Muhammad.” This wording is well known among both the Shīʿah and the Sunnah.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1