As the sishtAs of today, amongst us, think, "If somebody proposes a new path/ philosophy, we should not accept it directly; it has to be checked whether it follows the tradition or sampradhAyam", so also
our AchAryAl has given utmost importance to our sampradhAyam through out His bhAshyams. He used to refer with great respect that 'this is the sampradhAyam; this is how the sampradhAyikars opine; hence this
alone is acceptable'.
It is said about Jesus Christ that, 'He hasn't come to eliminate the earlier religions; but only to make them complete'. Even AchAryAl had come only to get the original-vaidIka-smArtha-sampradhAyam back on its rails rather firmly and He didn't establish anything new. He would consider referring to His as 'Sankara sampradhAyam' as an 'apachAram' itself!
He has established even His final conclusion - advaita - only based on the pramAna of texts, handed down by the tradition, such as upanishads, gItA and bhrama sUtras and wouldn't have mentioned any where anything as His own, even by mistake.
That is why, the followers of our AchAryAl did not get any new name; the title 'smArthAs' got stuck.
Before the advent of srI RamAnujar and srI MadhwA, even among smArthAs, a set existed which considered Vishnu as their 'ishta dEvathA'.
There is a funny aspect to this (idhilE oru vEdikkai): There were people among smArthAs who didn't have the [*spiritual*] maturity to engage in panchAyathana pUjA with special upAsanA for their ishta dEvathA and without indulging in 'para dEvathA nindhA', as advocated by AchAryAl. That is, they considered only their favourite deity as the primordial Godhead though at the philosophical level they seemed to have accepted advaitA.
Among such people, even the hardcore/hyper (thIvira) Vaishnavaites remained as advaitins. How [*I*] got to know this is through a fact, hitherto unknown to you, which I'll share with you [*now*].
There are few, who hail from such an advaitic-vaishnava-paramparA, exist even today. They hold only me as their AchAryA. If [*you*] ask about their siddhAntham, [*they'll say that*] it is advaitA only. They won't wear the nAmam. They won't wear vibhUti either, as it is considered to be associated with Siva. Rejecting both the nAmam and vibhUti, they wear only the 'gOpi chandan'.
Wearing neither the 'vada kalai nAmam' nor the 'then kalai nAmam', there is a sect, referred to as "kItru nAmak kArargal" [*those wearing kItru nAmam*], which continues to be advaitins. However, these advaitins are hardcore (vIra) vaishnavaites, when compared to those who hail from srI Ramanuja sampradhAyam. Even in the present days, when the RamAnuja-vaishnavites visit Siva temples, these advaita-vaishnavites, who consider me as their AchAryA, never enter a Siva temple!
I pointed out to that Japanese professor, "It follows from this itself that, what a great mistake it is to consider me a saivaite".
Similarly, there are advaitins even among the vIra-saivaites. Though they have accepted the advaita of AchAryAl as a philosophy, they could not take His advocated equanimity [*towards all vEdic deities*] at the upAsanA (theological) level and had ended up holding the exclusive superioriy of either Vishnu or Siva.