I feel the video's argument as to the relevance of fighting game terminology in Smash compared to other, traditional fighters basically ends this debate.
I'd also like to point out how bizarre it is that fighting games are pretty much the only genre that doesn't have at least two distinct variations on the core theme. Without getting into the minefield that is WRPGs and JRPGs, you can look at the FPS and take away that it can be streamlined, lightning fast action like Quake or UT, or much more precise, "hardcore" take like CoD or CounterStrike (forgive me for comparing the two, it's just in service of the argument). Again, in racers, there's arcade-y stuff like Mario Kart, Burnout and DiRT and there's more simulation-based games like Forza, GT and iRacing.
Genres tend to have sub-genres, of course, but for the most part everyone will concede that Pikmin and Starcraft are the same genre, as are Micro Machines and Need for Speed, and hell, even Portal and Tetris would be considered genre stablemates by most if you asked them to consider it. So why is the fighting genre so damn narrow? Answer: it isn't, obviously. Street Fighter, Guilty Gear, Soul Calibur, Power Stone, that fucking construction vehicle game and, yes, Smash, are all the same genre.