Thankfully, the government still exists and also works to solve problems in parallel with charities. I don't advocate that Buffett takes charge of the world. It's also nice that Buffett doesn't give all his money to one charity- he distributes it across multiple charities that (I assume) he deems as "effective enough to be worth taking a risk on," which is about as good of a guess as anybody has on anything. And even if his motives are more personal, the motivation of earning the autonomy to act for your own best interest is one of the primary motivations that causes people to do hard work that both earns money and (often, but not always, but I figure more often than not) makes the world a better place to live.

At a certain level, any decision made by any individual with autonomy is anti-democratic. The greater that person's autonomy, the larger the effects, and for billionaires those effects are large. They aren't always "better at picking" than the government, but having diversifying our methods of problem solving yields better results. The world would not get eradicated of polio as quickly if the Bill and Melinda Gates foundation didn't work on it. The Against Malaria Foundation does a more cost effective job of preventing malaria than just about any organization on the planet. Habitat for Humanity completes an amazing mission of building affordable housing, teaching skills to interested volunteers, and helping people invest in their community all at the same time.

If government is the only tool in the toolbox, you'll only hammer the nails that governments can hammer.