I don't think British people ignore the BBC. The problem with the US, because there is no publicly funded press, is that news in the public interest, i.e., news that a democratic electorate needs in order to have a legitimate democracy, is simply not disseminated. There are some private non-profit entities (NPR, PBS, Pacifa) that are close facsimiles, but they are either too biased in favor of their private donors (and hence not adequately covering the public interest) or are too resource-deprived to adequately disseminate necessary information.
It's got to be structured in a way that prevents that, i.e., institutionally given enough independence from the government. But this is a far lesser concern in an authentic democracy. In non-democratic societies, the state press is obviously horrible, but that's not a function of their being publicly funded, it's a function of the non-democratic nature of the government. Regardless, the best bulwark against a publicly funded media supporting those already in power is an egalitarian society with minimal wealth and income inequalities. Indeed, that's the best bulwark even against an undemocratic or abusive government. Democracies necessarily converge towards egalitarianism. If a particular society isn't converging, you're probably not looking at a democracy.
As to private media "expressing an opinion" that differs from mine. We're not talking about opinions. We're talking about news. Objective and unsentimental (and unsensational) reporting of what is happening in the society, in the government, and in the world. Private entities, particularly of the for-profit corporate variety, are incapable of doing that. Indeed, it would be illegal for them to under current law that imposes fiduciary duties on corporations to maximize shareholder returns.