I've actually tried to do stuff like this before. Offers of skilled assistance from individuals seem to be refused or ignored. You're welcome to fill some soup bowls, though!
The way I would approach it would be to attend a board meeting or email the Managing Director of the charity directly. Folks at the retail level of soup kitchens are really busy and deal with difficult people all day long. I think if you use a top-down sales technique to donate your experience you'll fare better. Yes, I believe you do, in fact, need to sell your donated time. Less sales really, and more positioning.
Source: I'm a marketer and I've failed at helping non-profits by starting too low on the totem pole. Conversely, I've also succeeded by starting at the top.
I'm not going to go around begging to be charitable, nor immerse myself in organizational politics that I don't put up with when I'm getting paid for it. I can't even begin to express how ridiculous such an idea sounds to me. At this point I'm happy to stay home and relax.
[1] https://www.kernel.org/doc/Documentation/SubmittingPatches
You're talking about doing something even more degrading than a typical interview process.
In the charity work that would translate to you offering your expertise to build something and when you don't have time or don't feel like the software/thing you contributed is cutting edge anymore.. then you quit. The nonprofit's "Linus" now has to decide what to do with your not-so-shiny system now that nobody knows how to maintain it and, to make it worse, a nonprofit is not a project like the Linux kernel with thousands of contributors, so it might be the system you contributed is now really critical and they can't throw it under the bus like Linus would do.
So that's one side of why nonprofits sometimes say no and you have to prove your way, not only to show your are capable (most volunteers really are) but that you are reliable and committed.
Of course each nonprofit will be in a different situation so even though they might perceive you as not reliable enough, they still think they might be able with you being gone one day so they say OK.
The politics side of business you don't like.. it doesn't exist because someone decide to call that thing a "company" or a "corporation".. it exists because the beings creating them are humans. It just so happens nonprofits are also run by humans.
It's hard to say no to someone offering their time and expertise, but unless it is focused as part of a larger, on-going plan, it may do more for the volunteer than the organization.
The first is that the food bank's mission is philanthropic, so any suggestion that "you should consider operating like a well-run restaurant to be more efficient" is usually dismissed.
Secondly, most food banks are under-resourced, so they're always "stuck in the weeds" by just trying to get everyone served, and that prevents them from taking a step back and thinking about how to make things better.
The third is simply human nature, people who have been doing things for a long time are resistant to change and perceive advice from outsiders as a threat. This is usually illustrated quite well in those "restaurant makeover" types of shows, when the celebrity chef and/or restaurant consultants try to talk sense into the struggling restauranteurs.
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).
This could be expanded into process engineering.
One particular non-profit I've had contact with is really bad at this. It verges on pure corruption at times.
Why receive a service for free from someone you don't know when you can pay a friend to pretend really hard that they're working?
See also Chesterton on the fence: http://stuartbuck.blogspot.com/2004/05/chesterton_14.html
I'm sure there must be similar organisations around the world - they are probably your best bet if you're looking to contribute your talents/time to assist a charity/non-profit.