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.
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.
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.
[1] https://www.kernel.org/doc/Documentation/SubmittingPatches
You're talking about doing something even more degrading than a typical interview process.
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.