Yes, that's exactly it. FPS, RTS, Fighting, and other genres all require different skillsets to be competitively viable. Although execution (the ability to aim and shoot accurately, for example) is important in an FPS, execution skill is not as vital as the greater strategy of the game. In a fighter, the intensely precise execution is not just an important factor but can be the sole determining factor between a miserable noob and a good (not great) player. Every game has a different balance of strategy and execution, and it happens that fighting games have, as far as I know, the highest ceiling of execution of any genre as a result of the intense nature of good combos. There is of course a decently high strategy ceiling in well-designed fighters, too.

Different genres can certainly be competitive online. In fact, all games but those with the highest execution ceiling are pretty viable competitive online games. Unfortunately, fighting games suffer massively from this, and, to me, fighters are emblematic of the once-great arcade culture in the US.

You may notice that I focus on the US in my comments, because Japan and East Asia in general still have arcades and, incidentally, tend to pump out some of the best new blood. As networks get better I really look forward to new generations of players in the US scene, but the arcade scene in Japan is really amazing. Unfortunately, unique to that culture too.