tl;dr: Youtube's content-ID system is the real problem, the DMCA is flawed and drastically needs updating, but would cover reviews under fair use.
I'm not going to claim to be a copyright expert either, but I am a first year law student that has discussed this topic with people that are. You're half right in the video: copyright as a concept was largely codified as it is today in the early 20th century, however, what we're dealing with now, on the internet, is the Digital Millenium Copyright Act which was largely passed in 1998 and amended later (though I can't remember when, probably 2004, but Wikipedia has failed me). But the DMCA isn't even the problem in this case, as all the reviews (and I imagine some of the other content) are covered EXPLICITLY by the fair use exception.
The problem in this case is Youtube's content ID system. Many companies that were cited as taking down content (see Riot Games, Deep Silver, etc.) have said they had nothing to do with it and will restore anything that people bring to them. What happened recently was they started applying this content ID system to PARTNERS in addition to random accounts. So people that made their living on Youtube were getting their videos flagged. That's why it's such a huge deal right now.
There are many problems with the DMCA, but the Youtube copyright scandal highlights one of them: it's damn near impossible to punish excessive and erroneous takedown notices. In order to do so, you have to take them to court, then prove, at trial, through a preponderance of the evidence that they did not have the copyright to that work and that they KNEW they didn't have the copyright to that work. You cannot take them to court for repeated violations of fair use.
On the direct matter at hand, my opinion is that Youtube needs to clean up their system. They should have a team dedicated to investigating copyright claims against partners at the very least, so they can catch fair use before this kind of stuff happens. Game and movie reviews should be protected and should be allowed to run clips of b-reel and trailers with impunity so long as they are doing it for the purposes of criticism.
The stickier point is let's players. On the one hand, it's arguble that their commentary is transformative and a large base of their audience comes to see them and not the games, however, an exhaustive (through the whole game, all options) let's play can definitely damage the saleability of a game and might exceed the scope of fair use. It's similar to reading an entire novel out and recording yourself doing it, what you're making is a bootleg audiobook and that definitely exceeds fair use. A good example would be pretty much any game by Telltale: if you run through Walking Dead or any of the Sam and Maxes, you reveal all the content to the player, there's very little gained by actually playing the game. Different, perhaps, from things like I Wanna be the Guy, Starcraft, and other competitive or challenge-driven titles where you can't get the experience solely from watching them. It's a legal question that hasn't been answered yet, and I think it's one that can only be answered inter-medium and not by over-arching legislation.
I think we might be in luck, though. The house recently passed the Innovation Act [http://www.washingtonpost.com/blogs/the-switch/wp/2013/12/05/the-house-votes-on-patent-reform-today-heres-what-you-need-to-know/] which comes down hard on patent trolls by making frivolous litigation very expensive. While this doesn't affect copyright at all, it indicates a trend that Congress is taking notice of the deficiencies in Intellectual Property Law and is doing something about it. We can only hope a followup to the DMCA (that isn't SOPA-esque) is what's next on the agenda.
