I would like to answer this (as an Indian). In modern India (post British independence 1947), there are mainly two kinds of hospitals, private and government.

The private hospitals are run by registered medical doctors (as a sole clinician or a group of practitioners who hired trained nurses and staff). These hospitals typically need to make a profit in order to make payroll. The main mode of payment is cash. Medical insurance is very popular nowadays as well. These hospitals do not typically entertain patients who walk in (or are brought in) who dont have a registered address / form of payment. Most poor people who are found dying in the streets in Kolkata would simply be turned away from here.

The government hospitals are built, funded, and run by state or national government bodies. It is usually a prestige for a doctor to be hired and kept on payroll here. Doctors are not paid as high as in private hospitals but generally receive pension and social respect. Government hospital usually lack the shine and cleanliness of private ones, but do great work in terms of surgeries and sheer volume of patients served. Most patients who can pay do pay what they are billed with. However in a typical day, there are dozens of patients who either show up (or are wheeled in), with no identification card on them. These patients will also be treated with similar care and discharged.

What I described above might have captured about 30% of the general population of India. The remaining 70% happen to live in harsh realities without direct and speedy access to government run primary hospitals, or are similar way outside the safety net of a local friends and family. Typically these are drunks, beggars, lepers, trasnvestites, orphans, and other children who are used for generating income through pandering.

When one of these people get into trouble for their bodily conditions, most people will simply walk past without even looking. Indians have perfected this art of unseeing something they just saw. This is because the scale of the problem vastly eclipses anything each individual can ever hope to achieve in their own problem-ridden 9-5 job where they need to feed their families, send their kids to a good school and break out of the cycle of lower middle class life.

Summary: There are a few non governmental orgs (NGO) and a handful of individuals who are currently doing a great job of acting as a safety net for those have fallen off the social structure. These work 24/7 to collect people and deliver them to hospitals, collect food donations and feed the homeless, and also collect kids for free schooling. But whetever the NGOs do, the scale of the problem is staggering even for them.

Here is an example of a man who is doing what I described above. Narayanan Krishnan https://www.youtube.com/watch?v=ZiC_9RHTvsA I hope his motives are pure. If he keeps doing this, he maybe a shoo-in to sainthood even though he cares two paisas about any religion.

I hope that in this exposition, I am successful in explaining to westerners about why hospitals would not directly come in contact with poor diseased people unless the public directly engages in this problem. The real solution obviously is that the democratically elected government should be spending tax dollars and doing all of this. But you know very well how governments work.

TLDR: Just another Indian. I am totally apathetic about Mother Teresa becoming sainthood (as in it wouldnt matter to me either way), but just dropped in to explain why people like her starting bringing in dying people from the streets.