All of those are current problems though. Signing media isn't hard, determining who is trustworthy and how to do identity is hard. This the HTTPS problem; you can show that the connection is secure (assuming people care enough to check in the first place), but you still have to check if the connection isn't actually to Target and is to Torget instead.

Signing only helps if we have a reliable way to verify identities out of band, and we don't, that's one of the reasons the other problems you mention still exist today. Today, how do you determine that an article is actually from the Washington Post? You check the website. It's not like verification of whether an article exists is hard today -- it's everything else around it that's hard.

If checking if an article is actually on a website is too hard for people, importing a signing key is also going to be too hard. If people are confused remembering the Washington Post's URL, they won't learn how to use a signing key to check identity. Now maybe a website could automate that and put some kind of verification badge next to trustworthy verified identities, but I'm skeptical because Facebook did try to do that with news sources and it was a disaster and a bunch of politicians accused them of censorship.

We have a lot of mechanisms for verifying identity and sources of information that aren't leveraged today, and I think the immediate question to ask about a chain of trust is "what's going to make this different from all of the other chains of trust that people are already ignoring today?"