I think the issue is more with motives. If people have money incoming from elsewhere then it shows, at least some, that they're not doing it because they just want a paycheck.
The trouble is that ultra-low-overhead charities can be completely ineffective. If a charity doesn't pay for computers, utilities, sorting machines, postage meters, etc. (or pays for only the bargain-basement minimum), then each hour of volunteered time is less effective than if there are sufficient facilities available. Overhead is, in reasonable quantities, a good thing.
Volunteer time is valuable, but it can help to have full-timers in administrative/organizational positions.
Of course, this observation is mostly based on larger charities, with assets and property and non-trivial operating costs. I could see how a tiny charity without these burdens could get along fine without.
I know of one charity that is pretty open and transparent about its operating procedures, scholarly findings and statistics, finances including employee compensation, and pretty much every other bit of data that doesn't compromise patient confidentiality (they operate a hospital in rural Nepal):
I totally agree, which is why I'm working on a startup to make charities more transparent and allowing them to make use of the resources they can get once they do this. (Andrew@brack.in if anyone has any ideas).