Hmm, if the government were to run a quadratic funding of charities thing, with only rather limited requirements for eligibility, perhaps that would somewhat alleviate the "undemocratic" complaint? (It would have to make it illegal to pay someone else to participate in your stead though.)
The rest of your post is based on the implicit assumpion that anything government run is inherently bureaucratic and slow and charities are instead simple and fast. But that's an orthogonal issue. Some level of bureaucracy is needed to prevent misuse, otherwise you have a rich guy pay their fines from their charity. Beyond that, there is no inherent reason for charities to be more efficient. You could make those governmnent-charities have local oversight, they could be organized very efficiently too and have the advantage of democratic legitimacy.
If only there was a massive logistics network for getting food everywhere we could use.
then you run into those who think , why should I give to charity because its the government's job to fix that...
and finally those who just don't care
charity exist for those want to focus their person effort to fix their world and should be encourage but never be required