Devatas are far more complex than people seem to comprehend. It is not enough to say it is a form of Devi, Vishnu or Shiva. Each rupam has a distinct personality and a scope of action and influence that is unique.
I was seeing a devotional Telugu movie Annamaya with Nagarjuna as a great devotee of Venkateshwara, better known as Tirupati Balaji. At the moment of darshan of the deva, his Shaiva forehead markings are shown to morph into the Vaishnava one. This might have irritated me at one time but it is psychologically true. Telugu land was indeed predominantly Shaiva at one time but the influence of Venkateshwara is everywhere now. Venkateshwara is a unique aspect of Vishnu not even completely an Agama form. In Andhra the Varaha and above all Narasimha devatas still go strong when all other parts of India they are near extinct. Even Tirupati was a Varaha devata at first as the sthala puranam explicitly concedes.
The temple was always super famous but not crammed in the current manner. My dad used to stroll in for darshan in the seventies. Somewhere in the 80s the devata went into a jana akashana and dhana akashana mode and since then it has been the wild numbers we all recognize. I went only once. I have no plans to return because I do not like to be locked in a cage while awaiting darshan. I am not blaming the temple for this. They have no choice. But this one and a half seconds darshan while being shoved aside with curses by arrogant temple staff is not for me. The murti is staggeringly powerful however and explains why the devata has become a transnational phenomenon.
Nataraja is also a devata who is unique. Yes it is Shiva but a Shiva that is practically known only to Tamizh worship. At one time the devata was an immensely radiating power. Chidambaram Nataraja was the kula nayaka of the imperial Cholas, and the source of their power and glory. Even in the Peryiya Koyil at Thanjavur , Raja Raja acknowledged in murals the source of his power. The roof was already gold when Raja Raja had the Chidambaram shrine depicted at Thanjavur. The temple was ancient and powerful before the Cholas too, but for them Nataraja gave special inspiration and ability to become the only Indian dynasty with a colossal overseas empire. Once that impetus was expended the devata changed his nature to become a more spiritual sort. There is a sthala purnaam legend that the daily expenses of the temple were met after a copper sheet they placed overnight near Bhairava used to turn into gold. About 400 years back that stopped happening which may mean royal patronage ran out or what it says. I am of course a hardcore Shaiva but of the Nataraja variety which is really different from all other rupams of Shiva. There are other Nataraja temples, but what is dancing in Chidambaram is really the only one of its kind in the entire Universe. Sarvam Shivamayam!