And why did they give up their agency to these shadowy media oligarchs...?

Answer: because they're stupid.

The ones who weren't stupid were impossible to herd to the polls, or at least a lot more difficult. As a result they were outnumbered. Any system that removes the influences you cite will leave the same stupid voters in place, ready to fall for the next con man who comes along.

The problem isn't the money. The problem is the power. I'm tired of giving stupid people so much power over my life.