I think the group that most reliably and effectively funds global health -- at least in terms of total $ -- would be the United Nations, or perhaps the Catholic Church, or otherwise one national government or another.
If you exclude "nations" then it does look to be the Church: "The Church operates more than 140,000 schools, 10,000 orphanages, 5,000 hospitals and some 16,000 other health clinics". Caritas, the relevant charitable umbrella organization, gives $2-4b per year on its own, and that's not including the many, many operations run by religious orders not under that umbrella, or by the hundreds of thousands of parishes around the world (most of which operate charitable operations of their own).
And yet, rationalists are totally happy criticizing the Catholic Church -- not that I'm complaining, but it seems a bit hypocritical.
I appreciate the good these organizations do, but I don't think that's the right measure of it. A person wouldn't in expectation serve global health better by becoming Catholic than by joining EA. That Catholicism is large isn't the same as them being effective at solving malaria. EA is tiny relative to the Church but still manages to support within an order of magnitude the funding you mentioned here, with exact numbers depending on how you count.
Similarly, it's not like government funding is an overlooked part of EA. Working on government and government aid programs is something EA talks about, high leverage areas like policy especially. If there's a more standard government role that an individual can take that has better outcomes than what EAs do, that would be an important update and I'd be interested in hearing it. But the criticism that EA is just not large enough is hard to action on, and more of a work in progress than a moral failing.
Rationalists and EAs spend far more time praising the Catholic Church and other religious groups than criticizing them - since they spend essentially no time criticizing them, and do occasionally praise them.
If you exclude "nations" then it does look to be the Church: "The Church operates more than 140,000 schools, 10,000 orphanages, 5,000 hospitals and some 16,000 other health clinics". Caritas, the relevant charitable umbrella organization, gives $2-4b per year on its own, and that's not including the many, many operations run by religious orders not under that umbrella, or by the hundreds of thousands of parishes around the world (most of which operate charitable operations of their own).
And yet, rationalists are totally happy criticizing the Catholic Church -- not that I'm complaining, but it seems a bit hypocritical.