Aadhaar is setup as a way of proof of identity and not proof of citizenship. I for one did not get an Aadhaar until a week ago! The only reason I had to was a company that I applied to, said that please use your full name as mentioned on your Aadhaar card.
India is a bureaucratic mess. And as UG Krishnamurti put it very succinctly, India is a failed country. As mentioned elsewhere in this thread, we desperately first need to focus on poverty first.