Yes, charities and NGOs like the American Cancer Society fund grants, too. But basic research is usually way outside their core competency, so teaming up with universities is their best option.

Companies team up with academic labs occasionally, and I think this is great.

But the government does already run its own labs and fund semi-independent research-only institutions. I'd like to see an expansion of these (mainly the latter), as well as Master's programs in science/tech, to help this current glut of graduate students transition to non-academic careers.