This is one of those platitudes that are designed to make you feel smart about yourself despite being pretty silly when taken seriously.
Consider this: The funding of the judiciary comes from the executive and legislative branches of government. Yet, funding is never brought up as an argument in discussions of whether the judiciary is independent.
The reason is of course threefold: there is a large societal consensus that the judiciary should be independent; the independence of the judiciary is established in constitutions; and political norms, as eroded as they may be in some parts of the West, aren't sufficiently eroded that politicians feel like they can play hardball over funding with the judiciary.
All these factors could be established for an independent, government funded media as well.