A good 2D game is a well executed game along with the subject preference of the gamer. I've seen good mechanics hammered by by poor design. I've seen good stories devoured by terrible game play. A good game is a well executed created game.
Past that I know gamers who just play for the game mechanics and skip as much story and as fast as possible. Then on the flip side I know some gamers who want less game play and more story. The idea of mixing gameplay and story is to broaden the audiance to increase potential sales.
Personally I like to divide the design down to difference of Sport and Story. Story is traditional to the merit of telling an engaging tell or lesson. "Games" at this point aren't games as they are a narrative experience. Sports however are defined as the players prowess and the challenge of getting better.
So what makes a good game. Just do one of them well. What sells well. Well WWE did it, blend the idea of challenge of a "sport" with an engagement of a story.
Anyways as a correction
--->Fighter
Street Fighter
Mortal Kombat
Samurai Showdown
Dead or Alive
Killer Instinct...
--->Brawler
TMNT: Turtles in Time
Streets of Rage
Final Fight
D&D Shadows over Mystara
Golden Axe....
these are not the same game genre. Fighters are a Sport as they are focued on the challenge of improvement, Brawlers tell a story of heroism.