You don't reduce distance by borrowing lots and lots of Tatsama words
The language of Musalman Indians - Urdu - is closer to Sanskrit genetically than all the languages you mention

Having said that, all the Southern tongues have perhaps more religious / H-cultural output in the past 1500 years, relative to Hindi in the same timeframe..
That's a different matter

So this is what emotionally trips Southern trads
Yes, your languages have more H-content relative to Hindi for the simple reason that much of H-developments have taken place in Deccan past 1.5K years
But it is also true your languages don't descend from Sanskrit

To give an analogy (rhetorical ofc)
My brother who was adopted by my parents, and who grew up with me, has a lot of cultural affinity to me and my family, unlike my own brother who separated from me at birth, and grew up some 3000 miles away