Yes, I'm aware of zero-knowledge proofs, but they aren't sufficient all by themselves.
> I imagine the problem is more economics and politics than technology.
Exactly so, yes. That doesn't make the problem any less serious. It also makes it very hard to put any real trust in technological solutions. You still have to trust that the solution provider is being honest, has implemented everything correctly, etc. I can't think of any entity that I have that much faith in.