It is very well possible that they would create other humanitarian organizations if the Church disappeared (I know they would, because I created a local charity back in the day), but would they be the same? Would they be as big? Would they be as diversified? Would they be able to stand up against teach unions? Look at the other organizations, how much money do they keep for admin purposes?
I maybe wrong, but let's look at the facts. There is still people that are jobless, hungry, homeless, without education, &c. where are the 'other' humanitarian organizations for these people that the Catholic Church hasn't covered? Is it because the Catholic Church steals all the charitable people? No, because you still have Red Cross, Shriners, Masons.
I think the reason is because the Church instills virtues in order to do it on a scale this big. Although, I still think the Church's charitable organizations are a little wimpy in America right now.