First of all, I am absolutely shocked that you accept the reality of the first sentence you're referring to. Frankly, that's a huge idea that even people who believe in limited government around here don't appreciate.
With that out of the way, I want to push back a bit on your idea that government has to be the champion of the poor or else no one will. I don't know what you're looking at that confirms government is the only eleemosynary actor we have, but I challenge you to cross reference those sources. Just as a point of contact, before we had this quagmire of a healthcare system, namely the employer mandated insurance system stretching back into the 20th century, the poor were taken care of through church funded hospitals. They weren't just dying left and right on the streets with no hand to help.
Now, you won't really have an argument out of me that the total contribution to the poor nationwide would be less than the amount government shovels out now, but you actually want that. It's advisable for charitable activity to be selective to the people that actually need the assistance, instead of blanketing money to people that either don't need it, or worse the people that developed a dependency to it.
Its also important to remember that because of our productivity increase we are even more capable of private charitable activity than in the past. (Though that's declined slightly over the last couple decades with our atrocious monetary policy decisions that wipe out the purchasing power of the middle class.)
Just on your own anecdotal basis, who do you know that truly despises the poor? Who do you know of that hasn't at least given a few bucks to salvation army or to a guy sitting on a street corner? Even if we were to be completely partisan hacks and say that the republicans just want to crush the poor into the ground, half of our country votes democrat and at least sympathizes with the less well off.