I sent this link to my sister, who is a director of a local charity. She's noticed this problem with overhead-fear as well. Every time her group gets funding or donations, there are strings attached. The strings are that none of the money can be spent on overhead.
It' so bad that her group has almost had to close shop on numerous occasions because she can't pay employees (they're overhead), buy equipment to needed to achieve their mandate (that's overhead), or let people know they exist (that's overhead as well). The group's very existence is considered overhead.