Aah, good ol' Extra Credits... If you ask me, the world would be a better place if more aspiring young game devs listened to what they have to say.
This episode in particular reminds me of all the student projects I've seen fail, because some innocent future game designer just
desperately wanted to tell that awesome story he had in mind, that would require a game world half the size of Skyrim, 20 animated characters and 10 hours of recorded dialogue.
Okay, more serious now. I think their advice here is a good rule of thumb. Focussing on the core of what your story wants to say and then developing a mechanic that works well with that sounds like an effective approach to me.
From what I've heard, I also think the developers of
Journey used pretty much this exact approach - with an amazing result.They first thought about the emotions they wanted the player to feel - say, love/friendship in this case - and then tried several gameplay approaches to convey them.
For example one of their first prototypes included a dragon that you had to protect your co-op partner from. But they found out that most players didn't feel like doing so and chose to save their own hide instead, which obviously didn't fit the message of the game at all. So they focussed on finding mechanics that would strengthen the bond between players (flying together by singing, energy recovery when standing close to each other...) and then found a story that matched these. In the end it didn't include a dragon anymore, but a mountain and a journey towards it, but the central message of the story remained the same.
Oh course there might be game projects where this approach may not work as well, like for example when you're making a game that goes with a movie and thus simply has to include certain characters, elements and actions. But in general I feel that when making games one should try to focus on what makes the medium "game" unique instead of merely trying to cram stories that would be better suited for book or film into this new medium.