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.
