You can really see how well such system works by observing USA right now.

Only way you could have any form of public financing of such endeavor without conflict of interest is to have multinational organization funded by every country.

Or you end up with BBC.

EDIT: to elaborate even further - you didn't even address the problem that ones designing this system would have to work against their own best interest. just wishy-washed that part away.

I'd say the US is a pretty shit example, given it's run by corporations right now, and lacks a judicial arm of the government that actually enforces the country's own laws. But to each and their own.

Again, with an open mind, go out and read about how publicly funded media works outside of the US (and UK, since you seemingly have a set mind about BBC too), and there is a whole rooster of different methods for funding these kind of things, yet letting them be independent. Some of these institutions are over 100 year old, yet still independent.

I'll leave it as an exercise for you to figure out how they made that work :)