My dad who fought in Korea always told us not to give the Red Cross the time of daySimilarly, my grandmother remembers her family being helped by the Salvation Army when she was a refugee in Europe, so I DO donate to that group. What these charities do or don't do will come back to them.
The Red Cross won't become obsolete any time soon, though, because they hit up businesses for money, they've got entrenched fundraising networks, and most people have a positive opinion of them, so politicians and other public figures won't speak out against them.
For what it's worth, the Red Cross won't recognize their Israeli counterparts. The Red Cross only recognizes 2 symbols, the red cross and the red crescent (arab). It will not recognize the red star of David, even though the Israeli group abides completely with the rules of the Red Cross and wants to be recognized by the International Red Cross. The International Red Cross goes along with the Arab boycott of Israel because they are afraid of losing support from the Arab donors.