This is a pretty problematic position to take. What you want almost certainly won’t happen, and it doesn’t happen even at the best funded media organizations. There’s no amount of effort they can put in to actually avoid any accusations of bias or inaccuracy from various political viewpoints. The internet is infested with a very vocal faction insisting the very existence of trans and nonbinary people is a falsehood, for example. Outside of clear issues like plagiarism, the validity of a story is in the eye of the beholder, a subjective call informed by the biases of the reader. What tends to happen instead is an enforcement from above of the political views, or at least acceptable window, by the editor in charge. You mention the New York Times as an example of an organization that does this right, but they’ve failed enormously on many occasions. For example, “alt-right” circles have organized successful bullying campaigns against writers for the NYT that have received validity and concern from the paper’s editors, however writers who helped spread lies that led to the invasion of Iraq in 2004 still write for the paper, unaccounted for.
Also, in the comments, you recommend Medium simply avoid promoting all “political” posts. This is just as problematic, as everything is political. The alt-right would use that policy to demonize the endorsement of LGBT authors and diversity advocacy, just as they use the mainstream media’s claims of objectivity to harass marginalized writers and silenced marginalized voices.
Now, fact checking? Sure, I am totally on board with that, in theory. I don’t see it happening, because it’s expensive and still requires putting your trust in the biases of your fact checkers, but at least it’s not nearly so dangerous as discouraging political views entirely.