the following questions have to answered on a factual basis
1. What exactly is the function of the matt has it changed in time and has it actually deviated due to the involvement of peetathibathis in politics etc.,
The functions of mutts may have evolved over time. The Shankara mutts were generally established as a monastic tradition, of the uttaramimansa vedanta specialisation kind. They read both shrutis and smritis, but are monks, renunciates, usually studied to be brahmavadins, but always aimed for moksha. They follow traditions that varied within, in 'small' ways (eg varying traditions within the dashanamis). That is how they also functioned in the 4 places where Sri Sankara established them (please note there were various ashrams present long before Sri Shankara established His 4 ones).
Over time, the Shankara mutts also seem to have started training priests of the purvamimansa schools, became involved in managing temples, etc. Perhaps they started veda paatshalas of purvamimansa schools to keep the traditions alive. But discounting this and talking only about the past, no one knows how and when the purvamimnsa ritualism school and uttara non-ritualism schools reconciled; or perhaps were not seperated in the first place.
I do not know if the purva mimnsa schools had their own establishments, which i think they might have had, as gurukuls where they were taught under a teacher, or perhaps they were taught by their father, but there is no clarity on how they became associated with mutts of monastic traditions that followed no ritualism.
One explanation is that since Sri Shankara won over Mandana Mishra and other purvamimansakas of his time, they became affiliated with the mutts. But it is probable, as it appears, that the purva ritualists gave up ritualism and took up the sanyasa or bhakti marga after being won over then (hard to know if there were grihasta priests present with the mutts then like the present purvamimansakas).
In terms of concepts professed, the purvamimansa and uttaramimansa may come across as non-reconciliable. Jaimini and Badrayan argued it out with their mimansasutra and brahmasutra respectively. The monks follow the non-ritualism of the uttara mimansa school and were considered brahmins since they sought brahman. The priests followed ritualism of the purva mimansa sutras and it is hard to say if they were considered brahmins at that time, since they believed in no brahman to merge into, at that time.
Some say the uttaramimansa texts were meant to be a continuation of the purvamimansa texts, and a man is supposed to use ritualism for a period and then he outgrows it as he moves on in his spiritual quest. It is hard to say if the monks and acharyas of the Shankara mutts follow this. Monks not associated with the Shankara mutts still do not profess purvamimansa ritualism (if you notice a good many sages in the pre-Shankara and ancient times were born out of wedlock, the rishis who sired them do not seem to have been purvamimansa followers) and they are considered vedantins or uttaramimansakas (followers of the upanishadic era perhaps).
There is also possible controversy in the following matters:
1) Non-Shankara mutts / ashrams allow women into sanyashrama (which Shankara mutts do not seem to admit as far as i know), something that can be considered controversial because it is supposedly not a uttaramimansa vedanta concept (disallowing women from sanyaashrama, it seems, is actually a concept from the dharmashastras or smritis).
2) Non-Shankara mutts / ashrams allow anyone from any jaati profession group seeking brahman under a brahmavadin or a guru (i dunno if the Shankara mutts allow the same, was told by someone they don't, which again seems to be a dharmashastra system).
3) Also am told the Kanchi mutt appears to be following the smriti or dharmashastra system of birth based jaati (which i hear is what purvamimansakas do).
Since things followed by the Kanchi mutt can come across more as following purvamimansaka and dharmashastra concepts, it is hard to say if it has remained as uttaramimsaka vedanta specialization school, as professed by Sri Adi Shankara, anymore.
But then if you see around, it is rare to find uttaramimansaka schools, pure vedanta, or pure advaitha anymore, not that they have ceased to exist, infact a good many seem to be thriving well (though they are not as popular as the purvamimansa schools) which is why one says the path to brahman has remained, still remains and shall always remain open to all.
In terms of what is followed by mutts, somewhere concepts mingled to co-exist and mutts perhaps just went by the changes in time (i suspect the co-mingling and subsequent changes happened in the aftermath of the mughal invasions).
2. Is it actually functioning as a centralized religious jurisdiction of the Hindus or just for the Iyers
I see that the Iyengars are not sensitized by this issue I am pretty sure there are a lot of Vishanvites in this forum
Only iyers i think. There is no centralized religious jurisdictions in hinduism. If diff people followed diff dharmashastras, concepts, beliefs, its tuf to think they can be centralized. i think the Shankara mutts are religious jurisdictions for smarthas.
As such abt hindusim, i think not being an organized religion is a huge advantage. It gives room, space and place for all kinds of schools, concepts, etc.
3. What exactly is the duty of the peetathibathi
and many more
i think the peethadhipathi conducts the daily prayers at the peetham, is a religious leader and guide for those associated with the peetham.
