It doesn't matter whether the fake stories actually had any effect on people's political leanings, because the media narrative is that they did. Ultimately, the mainstream media are in denial. Every single media organisation backed Clinton, including ones that had never backed a presidential candidate before, they published articles pointing to this as reason to vote for her whilst insisting that it was ridiculous to believe they could in any way be biased towards her. They were absolutely certain she would win, mocking those who though otherwise. They were wrong. Rather than look inwards, they found a scapegoat.
That's why they don't care about the 130,000 retweets for a fake claim that people were chanting "We hate Muslims, we hate blacks, we want our great country back" at a Trump rally from a known British hoaxer, or the quarter of a million for a photoshop of Trump's parents in KKK outfits, or any of the other fakes that don't fit the narrative.
As a liberal; I completely agree with what you say about the media organizations, and I'll add to that, some of the more-respected leftwing media; which used to be reliable, went into a phase of spewing absolute garbage. I found it disquieting, because it came about gradually, and I realized it suddenly, after the fact (about 6 weeks ago), that they were really playing fast-and-loose with standards of credibility on some of the stories they were posting.
Then, I basically kind of started to "shut down", because I felt that there was really no source that was trustworthy.
Around the same time, I saw some previously neutral forums, go "hard-Clinton". To the point where there was obvious brigading (downvoting and banning dissenters) going on.
On the other hand: There was copious material available, of direct quotes straight out of Trump's mouth. Live TV footage at rallies. Live TV footage at debates. Where he said some things that were not just; "oh, he's a bit careless and doesn't know what that's implying." He said some things that were very hard to interpret in any way other than how he said them. There's also the actual wording of actual policy positions on his own web site.
I think this is what is meant by the assertion that "false stories from the left are not nearly as big a problem".
It's true that there's still some crazy stuff coming from the left. (A lot of the news organization sites breathlessly speculating about who Trump is appointing, before he's actually made any appointments - and turning them into outrageous scandals).
But a lot of the negative things coming out about Trump are absolute facts that you can't sugarcoat, because he said them while being recorded.
The thing I accept, possibly wrongly, which others have stated, is that many people voted against Hillary and there was not much that could have gotten in their way.
Given Trump's past political opinions contrasted with his campaign claims, people who voted against Hillary often times took what Trump said more metaphorically than literally, whereas they took Hillary literally probably because she asked people to take her literally (that other guy flip flops, I don't). They saw Hillary as an ideologue and Trump as more of a pragmatist, something Obama apparently believes as well.
Trump is appalling. But media went from reporting that Trump was horrible, to cheerleading for Hillary. Some of them went further, and tried to be on the playing field on Hillary's team. That's a problem, and it's not justified by how horrible Trump is.
My favorite plausible one was the one where supposedly back in the 90s Trump said "he'd run as a Repub because they're the most gullible". I found it amusing that Hollywood actors/actresses tweeted it in a non-jokey way.
And yes, it is telling all their examples are about ones which targeted Hillary.
That's why they don't care about the 130,000 retweets for a fake claim that people were chanting "We hate Muslims, we hate blacks, we want our great country back" at a Trump rally from a known British hoaxer, or the quarter of a million for a photoshop of Trump's parents in KKK outfits, or any of the other fakes that don't fit the narrative.