Americans can't differentiate entertainment from journalism. Also Trump calls anything that isn't state-sponsored propaganda lies, even stuff that is entirely backed up by fact, corroborating evidence, or in some cases even video/photographic evidence. It was literally the firs thing his administration did when he marched Spicer out to complain about crowd sizes, when the truth was evident. You all follow him into the indefensible positions and just attack the source and not the content or facts. Once your base believes your blatant lies, you can then call anything into question.
The right want to rail against anonymous sources, while Trump himself is one of the more renowned anonymous sources. He loves to talk to the press, and has been doing it for years and you think he suddenly stops when he gets into the oval office? He loves Haberman and is probably the source for a lot of her "a senior official" stories. It's his thing...he floats ideas out there anonymously...when they flop with his base he calls them lies.
What would you do as a reporter when Trump, Kelly, Mattis, Kushner, Pompeo, Barr etc. call you up and give you a story scoop but then tell you to not quote them, but use them as an anonymous source? Literally every one of them do it, because they all have their PFA within the administration and are jockeying for power, and use the press to move the ball behind the scenes. It's an integral part of the process, and yes it is up to the journalist to sift through the drivel, but in the end if the journalist wasn't there they have to rely on their sources and how reputable they are.
How do you think journalistic integrity could be fixed in the eyes of the right and there be actual critical reporting of their actions and policy and still be viewed positively? It's entirely not possible...the right is to the point that even Fox News is seen as being too left.