As I have said many times before. the linguists seem to disagree.The separation between Kannada & Tamil (& Tulu) or between proto Tamil-Kannada & Telugu is too far back as compared to a more recent split between Malayalam & Tamil.
To imagine that Tamil would have been same as you suggest & other languages change that drastically is beyond common logic. Where are the Tamil inscriptions deep in the homeland of the Telugu & Kannada speakers if their language changed later. Other than the odd specimen of the script in areas close to Tamil lands, there aren't that many. There is no evidence, archaeological or linguistic to suggest remotely that people in the places where they now speak Kannada & Telugu were wholesale speakers of "Tamil". A couple of interesting links.
EPIGRAPHY - Inscriptions in Tamil Script
Kannada inscriptions - Wikipedia, the free encyclopedia
As can be seen, there is no evidence of a massive gap between the earliest inscriptrions
( a few hundred years are not enough to both create a language & also form a script), the split must have occurred much earlier. There is no other explanation that makes sense.
This particular line is extremely interesting. If, as you suggest, the neighboring languages originated in Tamil, why did they chose to use Sanskrit & Prakrit & not Tamil. When another language script is used, it is the language of the region & not a supposed
"origin language". The evidence seems pretty clear except to those who prefer to not see it.