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?

Hypothetically would you rather have:

1. Someone making $100k who is able to bring in and distribute $3m in donations per year

2. Someone making $300k who is able to bring in and distribute $5m in donations per year

There's real value to having someone who knows what they're doing in the top position, and that won't always be the person willing to take a pay cut.

  >  someone who cares deeply enough to take a pay cut to work there. 

What makes you think this isn't the case? In my limited experience, it is very much the case.

This makes a terrible assertion about what's in people's hearts, something you cannot know and ought not to make accusations about.