The Kings of TN had a Sanskrit Name for the simple reason that their territory was much bigger covering multilingual areas. Even the second Pandyas ( Kadunkon line) who ruled only South TN and some times upto Then Pennai also had Sanskrit names as they had relationship with all over the country.
It is easier to understand Rajaraja than Arunmozhi. Even Rajaraja could be pronounced only by Indians and Far East. Chinese ended up calling him Lotsa Lotsa.
I gave ref of Karikalan doing Velvis.
பாடப்பட்டோன்: சோழன் கரிகாற் பெருவளத்தான். திணை: பொதுவியல். துறை: கையறுநிலை. (Puram 224)
இந்தப் பாடலில் கரிகால் சோழன் தன் மனைவியரோடு வேத விதிப்படி யுப ஸ்தம்பம் அமைத்து வேள்வி செய்தது கூறப் படுகிறது.
All across the country the Sadangu part was in Sanskrit. The customs of marriage, Shrard etc were uniform and in Sanskrit. Whether a south Indian dose shrard in Gaya or a North Indian dose in Rameswaram - It was and is common and in Sanskrit.
Sadangu is something should be common across the religion and all religions follow this.
Where as vazhipadu is a different and all states have this in their mothertongue. Not only in TN, all states have like Hanuman chalisa, marathi abangs, Telugu keerthanas, Kanakadasa krithis etc.
The so called Sanskitisers have equally contributed to Tamil Litrature and Tamil Vazhipadu.
Sambandar who confirmed - Enadhurai thanadhurai" - My word is his words. Sundarar who had Lord as his friend or Periyalwar who married his daughter to the lord or Abhirama bhattar who brought moon on Amavasya day or Oothukadu Kavi who made Krishna dance to his Tamil tunes to Mahakavi Bharathiar, Papanasam Sivan and the list is long.
Tamila or for that matter all Indian are clear that for Sadangus - It is Sanskrit and for Prayers ( Vazhipadu ) it is Tamil.