I think the comparison to Red Cross/Red Crescent is not quite up to the point: my impression is that, for some things that Europeans see as a matter to be regulated (i.e., force health insurances to include "expensive" patients in their coverage, have the state provide subsidies) people in the US rely on local communities (churches, clubs) or their social network. Hence the notion of "paying it forward": by giving to your community, the community acts as an insurance and is likely to reciprocate if or when you are in a bad position.

This is why, for many people, it's a big deal to be an atheist: if the main community for you is the church, you're not just foregoing religious and moral support, but also the material kind.

It's certainly an understandable reaction to point out that the problem is bigger than one person on giveitforward.org, and that the US health insurance system with insurers being given cheap excuses like coverage ending with your employment contract (rendering your health coverage useless whenever you have a severe disease) or allowing insurers to retroactively (sometimes by 10 or 20 years) invoke a preexisting-condition clause means that many more people have to beg the last months of subsistence off of their friends, family, and community. Or just sit in a corner and die, if they're not well-connected socially.