Without Dunant, the Red Cross wouldn't exist. Please think about this. You're right, a lot of the tasks they fulfil today would have been picked up by other NGO's (also, I've learned the RC is not technically an NGO nor governmentally bound, rather something in the middle, but blah technical terms), and that's correct when you think about the humanitarian aid which the Red Cross is probably most known for.
Yet the main reason Henry (my love) started this organisation, was to protect wounded soldiers, and the First Geneva Convention was signed in 1864. The four Geneva Conventions are part of what is called International Humanitarian Law, which regulates (or tries it best to regulate) war, and is extremely important.
For me personally, the Red Cross has been a place to grow up, to find friends, to learn so much about myself and the world and history, to flirt and play and get work experience and cry and create some kind of meaning in my life. Sometimes though, it really strikes me how massive this organisation is, how much
I could go on about the Red Cross principles (there are seven) and how I think this organisation is on a much higher level than many, but let's save that for a rainy day, shall we? Before this post turns into a TL;DR, if it hasn't already.