It is difficult to disentangle the different roots of what is now known as Śrīvaiṣṇavism, since this term is usually the label attributed to the religious counterpart of the philosophical-theological school of Viśiṣṭādvaita Vedānta. However, Vaiṣṇavism was apparently an important presence in South India well before the beginning of the philosophical enterprise of what later became its philosophical counterpart, namely Viśiṣṭādvaita Vedānta.
Accordingly, each study on the making of Śrīvaiṣṇavism needs to take into account several distinct moments in the development of Śrīvaiṣṇavism: on the one hand its pre-Viśiṣṭādvaita Vedānta Tamil existence and on the other its theological evolution within Sanskrit (or Sanskritised) philosophy and as Viśiṣṭādvaita Vedānta. The main referents for the first phase need to be the Āḻvārs, the poet-saints whose work has been preserved in the Divyaprabandham.
For the second phase, one needs to deal with the formation of a philosophical school in the work of the predecessors of the first teachers of Viśiṣṭādvaita Vedānta, namely Nāthamuni and Yāmuna and then with Viśiṣṭādvaita Vedānta as found in the work of Rāmānuja, Śrī Sudarśana Sūri and Veṅkaṭanātha (and other scholars).
The necessary connection of these two phases has been stressed again and again by the authors of the second phase, who —from a certain moment of time onwards (possibly from the time of Veṅkaṭanātha)— enumerated the Āḻvārs among their teachers and forerunners and even tried to connect their tradition directly with that of the Āḻvārs. Neevel (Neevel 1977, p. 11) has for instance convincingly argued that the life-spans of the most important authors of the philosophical formation of a Viśiṣṭādvaita school have been prolonged in order to connect them to each other and to link the first of them, Nāthamuni, to the Āḻvārs, although this led to the attribution of a lifespan of 330 to 340 years to him.
To sum up, Śrīvaiṣṇavism became what is known to us as Śrīvaiṣṇavism through the heritage of the Āḻvārs, the ritual contribution of the Pāñcarātra Saṃhitās and the philosophical contribution of Nāthamuni and Yāmuna first and of the Vedāntic reshaping by Rāmānuja thereafter. Veṅkaṭanātha might have been the attraction pole completing the puzzle of all these loose elements.
(cross-posted on my personal blog)