The majority of their income comes from government subsidies and from charitable donations. At some universities, over time, that's built up into endowments which support professors.
That's very much the definition of "live on charity."
Charitable donations from wealthy people who, for some reason or another, feel indebted to their alma mater. How successful would these people be without the education they received from their professors?
Educators produce enormous externalities that their students capture and build their lives around. To turn around and say they "live on charity and lecture us on money" is the height of ingratitude.
Many charities produce enormous externalities. That's why people donate to them.
I think OP's point was that those organizations don't necessarily have a good grasp of the reality those of us bound by market forces live under. That's something I've certainly seen from both universities and other ivory towers.
For the past many decades, I've donated to organizations like the FSF, mostly to allow people like Richard Stallman to be funded to do what they want. That gives those people great freedom to do the right thing, and it's well worth the money. That doesn't take away from the reality that those people are now in no position to evaluate the choices I make, without the luxury of that same sort of support.
Hence terms like 'ivory tower.' Yes, we should provide artists, scholars, and others with charity support, but yes, we should accept that distorts their worldview too.