I think it's because they don't have any way to vet your credentials. You say you're an efficiency expert and offer to re-work the food bank's delivery process, what if you're incompetent and make things worse? They don't have the time or resources to fix people's fuck ups. The food bank has a pretty solid level of confidence in your ability to pack boxes though.
How would they accomplish anything, then? At its simplest, somebody has to deal with the money, how do they vet that person?
In theory, all the ways an employer vets someone are open to them. Review a resumé, contact references, ask other people who might know the reputation.
If this is really what's stopping them, then it might be time for someone more motivated than I to set up a non-profit to deal specifically with this sort of problem for other non-profits.
In the area where I live and work, there's a charity called Activate Good (http://www.activategood.org/) which is designed to place willing volunteers with nonprofits who need help. It's really quite cool for those of us who want to arrange for teams of people to do community service but don't have a lot of time to research worthy causes. All I have to do is tell them about how many people will be there, and they'll find something for us to help with.
They also look for specific jobs (like marketing, copywriting, IT help, etc).
