I tried this, too and realized similar things. I then decided that I do not really care (at least not that much) about understanding which way each source wants to spin things. I instead want information about the world to form my own opinions.

I started reading international news. That is, focus on publications outside country X when reading about X.

Reports from Sweden, Korea, Russia and UK (thanks google translate!) translated into English, awkward wording and all, plus a minimal dose of CNN and Fox works better for me than a mix of American media. Just my 2c.

I might even wrap it up as a convenient page or app.