"Most words are either borrowed from Tamil or Telugu or Sanskrit"
The exact and no point for refusing your factual words. Asal Kongas why still bark when they are in a position to see such a pathetic siuation for self.
Emergence of telugu from earlier day fine tamil is least to base. Bze we need again to go back more to sindi, bramhi where tamil shares common base with them. (Moreover today these two languages sindi, bramhi are much influenced and conquered by foreign languages.) But kannada and malayalam were clearly derived out from Tamil. And when it had faction-ed and formed its own identity it is not easy for them to accept the roots. You cant expect the same bengali-oriya rship here as the other side here are apes.