The issue is that your average American (including Trump voters) don't want to eloquently say all of what you said, just to denounce CNN. It's much easier to call it "fake news" to let everyone know that you don't like, or don't watch CNN -- without having to say much more than that.

Now, by calling it "fake," people then associate the organization with "incorrect facts" or "propaganda". CNN holds it's own agenda and does get some facts wrong from time to time, but most of the news is fairly accurate. As you mentioned, it's more for entertainment rather than to inform. The worse part is that most who would call CNN "fake news" are probably avid watchers of other entertainment-news outlets such as Fox. And the opposite can be said about CNN viewers -- who call Fox news the worst news outlet on cable, while being blind to the similarities between the two networks.

I like your opinion of the Washington Post, although I do not think we should ever limit our readings to our favorite outlet. Read the NYT, watch Fox news, subscribe to Reason, etc. Trying to filter through all of the media is much better than tuning out the ones you don't like.