Besides I am not sure bringing nationalism into charity is classy or even logical - given Bill Gates is spending his own money as opposed to tax payer one or national funds.
India is in a unique position to solve these issues, and hopefully some of those solutions can be applied to other places in the world, including America.
Yes, there are worse places, and while I've never traveled to Mumbai, I do know people who live there, and there are plenty of people with money there, and it's not all slums.
There are rich everywhere, and there are poor everywhere. The reasons for poverty are diverse. It's not: America is rich and India is poor as he seems to show.
But what do I know...
Look what a long letter I write... I would suggest that the only way to help the people of the slums is for the rest of the people of India to want to help the people of the slums. There has to be a mindset that the majority of people in India adhere to, namely, that poverty is not acceptable, and that the factors that enhance poverty are to be remedied, even if this means pushing aside long-held traditions and ways of doing things.
Which is a tall order, even for Bill Gates.
So instead he goes after specific problems, such as polio, and it's all good, don't take me wrong, but, huh, it's not nearly as impressive in the grand scheme of things as it's made out to be.
Gates is working in India on education, on environmental safety, on sanitation (seriously - I've heard talk repeatedly that his number one goal right now is a better toilet / sanitary mechanism) - these are things that will directly advantage India in the long run. As for the Indian government... well, the road to hell has been paved with good and not so good intentions - suffice to say that has not always been the goal.
FYI separate topic, but the UID is a success story by any standard (its the world's largest biometric database by some distance) - and enables a reduction in the wastage in the traditional channels you identified (aka, the politician cut).
I think he's doing good, true, better than nothing.
Just take a trip to Mumbai and visit the slums there.