None of this giving goes to government organizations, although of course some nonprofits get federal or state funding (probably a small %). But the point is that millions of Americans find a way to help the underprivileged apolitically

Contrary to what you might expect from the name, a "nonprofit public charity" does not need to do anything to help the underprivileged. For instance, if the exempt purpose is "religious", and the organization does nothing but cater to the religious needs of its members and does nothing to help the "underprivileged", it is still a legitimate public charity.