As someone who is paid to review games, and has done that as a job for multiple outlets across my lifetime, I think you should have to have a base level of competence at every genre in order to be in this industry. I'm not saying you have to be great, or even good—but you should be able to pick up a controller for any game and be able to play it. That is a basic skill you should have if you're going to play games for a living.
Beyond that, specialize in what you do, and make sure the right people are getting the right games. I'm not sure the story behind how this video game together, but I would personally be embarrassed to have this video up on my website—and I say that knowing and being friendly with some of the Polygon folks. There's just no way this video should ever have been cleared for approval.
I don't care if it's just showing off the beginning of the game or not, have someone who can properly show off the game, and not make your outlet look bad.
Edit: The above Gran Turismo TGS video may not be totally fair, though. I'm not great at hardcore racing games, but I can play them—and yet I might look like that if I sat down to try a racing demo with a wheel as part of the set-up. If you aren't used to wheels, they can be super tricky to get the hang of. And I play all racing games in third person, so come at me.