I can confirm the OP's experience albeit in a different country (Mexico). In some regions, I haven't been in a single street taxi with working seat belts. Indeed, I've been in some private cars without them as well, or with more physical space for passengers than there were seat belts available.
Uber vehicles, though, have always provided those safety features. Some drivers work as both street taxi and Uber drivers (as they frequently cross into or live in regions without Uber but drive people to places that have it), so that quality assurance can trickle down in some cases. It honestly goes beyond seat belts though; a Uber car is more likely to have AC, electric windows, etc. than your average taxi, even in ridiculously hot parts of the country.
Maybe because it was in Chennai, a southern state and not in Mumbai or Delhi.
That sounds so weird to me... What do you mean cars don't have seat belts?!