For example, one charity I like [0] runs a school and the administrators don't take any income from it.
Some non-profits only pay salaries of employees, but spend the vast majority of their time and money on their fundraising events, which are just ways for rich people to throw parties for free. In that case, they'd still pass your test.
People who work at non-profits deserve to be paid a similar salary to what they'd be paid in the private sector. The important part is that a non-profit should be using the majority of their funds on their mission (and that may require the majority of the funds going to the workers, depending on the mission).