It is obvious that a name based on mythology, for instance, Europe from the myth of Europa, need not be mythological itself, and is a real name. So, too, in the case of Bharata Varsha, or Bharatavarsha.
Regarding the Indian use of Dravida, it is meaningless to state that "Indian scriptures always referred Dravida as a region extended to Chola, Chera, and Pandya." What does that mean?
That there were no exceptions? That could be because of the etymology of the word in Sanskrit.
That the region ruled by Chola, Chera and Pandya was collectively known as Dravida? That there was no other meaning of Dravida, other than this, a region in India? In Sanskrit, there are two words in use, transliterated into English as Dravida; one word is Dravida, Tamil, and the other is Draavida, of the Tamils. The second may be used of a region; surely not the first?
Again, the erroneous racist theories of the nineteenth century counterposed a Dravidian race to the Aryan race, and both are dismissed today as not fit for rational consideration, because of their grotesque misuse and horrible consequences of racist theories, with a significant focus on the use of the word 'Aryan' as the name of a race in real life. It is widely accepted today that Dravidian can mean the family of languages grouped as that, ie, Tamil, Malayalam, Telugu and Kannada, Tulu. This is certainly an improvement over the narrow sense in which Dravida was used in Sanskrit, for instance, and expands the scope of the word to those very different languages used in south India. The word is also concurrently used in several other contexts; none of those necessarily is exclusive.
Coming to the very different uses of the word Aryan and its use in a combination as Indo-Aryan, it is not clear what is objectionable about the word Aryan being used in combined form to name a language in the language tree descended from PIE. Surely it is not rocket science to understand that a word used in one context, for instance, Latin, might refer to something entirely different in combination, such as Latin America.