What he says is bad is shifting the decision to the wealthy instead of the public.
Now, if the public is deciding by choosing where to put their own funding, in the form of donations, that makes more sense. But in that sense, the millionaires and billionaires creating or donating to charitable foundations are doing the same thing: deciding where to put their own funding.
It's his money, so it's his choice where it goes. Beyond that, this is just an argument for higher taxes to take that choice away. If you want to argue against allowing people to have and direct the wealth they've created, then make that argument, rather than assuming it as a premise. Once you've assumed that as a premise, then it starts to sound obvious to deemphasize charity and emphasize government programs; that step isn't the interesting part of the argument. As it turns out, we have multiple countries in the world, with different sensibilities about the relative value and efficacy of public and private work, among many things. The US model, evaluated by European sensibilities, is broken in many ways; the European model, evaluated by US sensibilities, is broken in many ways. Neither of those is as interesting or relevant for evaluating potential changes as a country's evaluation of its own models by its own sensibilities.
Personally, I'd be thrilled to have significantly less of my income directed by government to the causes they care about; I currently donate a significant fraction of my annual income to charity, and if I had the money back that currently goes to a pile of things I mostly don't care about, I'd love to donate even more to charity. I know for a fact that it'd do more good. Changing that would require changing what government is spending the money on, and I'm far more confident that I can change the course of a charitable organization (or choose a better one) than change the course of government.
As it stands right now, I think I could do more good with a pile of cash by setting it on fire than by giving it to government; additional money in government hands has negative utility, and may be used to hurt me and people I care about.
As a random thought experiment, I've often wondered how it would work if charitable donations produced a deduction directly from your tax burden, rather than from your taxable income. In other words, every dollar you donate comes straight off your taxes. That would produce some interesting results.
How is that any different from the comment you're replying to? If the government is running something, it's a government ("public") program rather than a private one.