According to my late paternal grandparents, who went through WWII, Korea, and Vietnam on active duty (and arguably the wives did as well), the American Red Cross had not-so-nice aspects to it that didn't get much press.
Perhaps national governments should support apolitical, non-religious volunteer nonprofits more, but I can also understand the need for nonprofits to survive (which may often entail charging small fees). Charging people for doughnuts who don't make much money to begin with, are first-responders, active-duty military, or individuals who just went through a disaster seems kind of uncool where I come from.
It's difficult to say how good they are today without first-hand and multiple accounts of experience. The available data shows they presently spend 3.5% ($104m) on admin and 6% ($177m) on fundraising. https://www.charitynavigator.org/index.cfm?bay=search.summar...
As an example, Feeding America, although they pay their CEO 16% more ($100k+), is a mostly better charity on paper.