I subscribe to the Elamo Dravidian hypothesis that the entire swathe of peoples from Tamil Nadu to Elam to Sumeria were related. I’d extend that to the agglutinative languages of the Kartvelian and Baswue language families
Nothing much had changed until the British arrived.
The possibility of Tamil roots in Harappa looked plausible to me as well, but the relationships between agglutinative languages is ever in flux. Most eastern Asian languages are agglutinative, but not necessarily particularly related to one another. And language comes and goes, and when it's gone, it's gone.
Harappan has been cracked a bunch of times, but none of the purported decipherments has caught on/held up under scrutiny. Most accept that the language (if any) has an agglutinative structure.
Sumerian is an isolate, with very little known loan vocabulary (they adopted the place names and river names, as by their own account, they came from elsewhere). That makes the problem worse, as the deviation in the changes in possible related languages can't be documented, what with Sumerian being the oldest known written language. Their relatives are entirely silent, probably permanently.
[snip] The ancient Harappan genome, sequenced and described in the journal Cell, was compared to the DNA of modern South Asians, revealing that the people of the IVC were the primary ancestors of most living Indians. Both modern South Asian DNA and the Harappan genome have a telltale mixture of ancient Iranian DNA and a smattering of Southeast Asian hunter-gatherer lineages. "Ancestry like that in the IVC individuals is the primary ancestry source in South Asia today," co-author David Reich, a geneticist at Harvard Medical School, said in a statement. "This finding ties people in South Asia today directly to the Indus Valley Civilization." [/snip]Rare Ancient DNA Provides Window Into a 5,000-Year-Old South Asian Civilization | Brian Handwerk | Smithsonian | September 5, 2019