You joke, but PBS is exactly that - when other countries adopt its programming, it's helping push Western/American culture. I grew up in Brazil and we had a local licensed version produced with Brazilian actors interspersed with dubbed original material.
My point, which is not a joke, is any media funded by the government will soon find itself pushing only the point of view of those in power in the government.
Absolutely not. You can make funding independent from government (see BBC's TV tax, for instance) and have a board of trustees that cannot be nominated by any government (more or less the same model as the SCOTUS, or make it so that the board can't be changed by more than 25% of its members over an 8 year period), and create clauses that ensure it represents various groups of society, in particular the underrepresented. While you are at it, make some bylaws that the entity can't change itself that ensure the board can't hijack it. Hire professionals using clear competency measures and give them tenure.
Would have to rename it to Propaganda Broadcasting System.