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You could do that on kickstarter. An audience exists there already.

Although it's mostly small investors/ backers there. You have thought more of the bigger investors, haven't you? Research is expensive indeed...

Unfortunately democracy doesn't do great in areas where technocrats are needed.

Um, a lot of research is, in goal, substance, and subject, completely incomprehensible to non-specialists. So in practice I see a lot of obstructions to this method :/

to a first approximation, what you’re proposing is that citizens pay their taxes, and then the state funds research at accredited non-profit institutes of higher learning.

it’s the best we have right now; it is not without drawbacks. You’re not going to escape wanting to reinvent the grant proposal system.