They do take a pay cut. There are no stock options at a non-profit. Salaries are lower than in the private sector. If you think non-profit CEOs are raking in millions, no wonder you think they are over paid! Don't worry, they are not making anywhere close to what a similar role in the private sector can earn them.
How much you care isn't as important as how effective you are at running the organization. That seems like a counter-productive optimization. Let's say we have two charities: one that is run by 'bleeding-heart' passionista's who live in near poverty themselves, or one run by master marketeers and logistic experts who collect sustainable and fair wages. Which do you donate to and why? Which one will actually solve the problems you want them to solve?