Can you give me some examples of things like complex numbers discovered in the private sector, by charities, or 'enthusiastic amateurs'?

Examples that fit Salemicus's narrative in this context aren't non-existent. For example, Fermat was a lawyer by profession and did math as a hobby in his free time. There are many similar examples prior to the 19th century or so. And some major charities have helped fund successful research- one sees a lot of this with a variety of diseases.