Having non-profit collecting crowd-funds for the solutions to the important scientific problems would be great. But I would imagine that for those most capable of solving these problems prizes are not essential, while grants that would allow them to conduct the research in the first place are.

So... Create capable non-profit with goal of crowd-funding important research.

The key innovation would be an investor (venture capitalist?) willing to give out grants themselves, in return for a share of the resulting X-prizes. Then funding the prizes would, through Market Magic™, produce grants.

My guess is except for the prizes with a big material resource need (any space one, for example), this will not make it more likely

Just see the Poincaré conjecture and how who solved it turned down the price

For most of the math prices the solution is a very big and inscrutable collection of mathematical theorems, lemmas, proofs, etc