Privatizing government institutions won't do anything to help, but it is nonetheless the case that the American government has a large amount of responsibility for what's going on. The reason why American companies cannot simply sell food to Gaza is because the government of Israel controls access to Gaza and the government of America supports them in this.

Without the American government's various anti-BDS laws, it would be much easier for American organizations to apply pressure to Israel to stop their genocide. Furthermore, without the American government providing support for Israel, they'd be in a much tighter spot and would face significantly stronger incentives to stop.