Of course in that scenario the BBC would be responsible as they're the ones publishing it. Who else is responsible for the content the BBC publishes? I say again, it's not the job of the press to act as PR people for individuals or organisations. That is clearly the case with your Everton example, as the Everton website is run by Everton, who employ Benitez, so it is most definitely PR, not journalism.
No, C-SPAN is a news network but it also acts a bit like Reuters or AP. They have a lot of video teams out there. They put videos on their wires, this is a video of a statement to the press. In this video, which was a 2-min clip, there was no opportunity to ask questions. Now, C-SPAN then should - and I don't know if they did but if they're a news network I'm sure they will do - have had people assessing what was said. Under separate cover. But posting the clip itself is not misinformation.
To use another football example. When a manager lies about a possible new signing in a news conference. Let's say Benitez says today: "No, we aren't signing Longstaff. We will not make a bid for the player, he does not interest Everton." That clip, cut to 10 seconds, would then be used across lots of different sites and broadcast agencies (the company I work for being among them, that's how they work in the vid teams).
Then let's say on Monday, Everton sign Longstaff. The companies that spread that clip are not in any way liable for the spread of misinformation. They have cut a clip of Benitez's quotes. There would almost certainly not be any inclusion of the journalist asking the question, or a follow up question, it would just be Benitez's quotes, as that's how viral videos work, by and large unless there is an instance were the publication in question wants to show that a certain question was asked.
And back to the C-SPAN vid, it was simply a video of the person making a statement to the press. In that video, no questions were asked, as it was the initial statement. Other videos will, almost certainly, have shown journalists asking follow up questions if it was a press conference setting. It was simply a representation of what was said, a coverage of an event.