There is some great conversation so I'll try not to ruin it with my 2 cents.
I grew up a Bullseye shooter, 25/50 yards, one hand, unsupported, 5 shot strings. If it was outside the black it might as well be a miss and if it's not an X...well, hope the other guys are having bad days too.
Then I get turned on to IDPA USPSA, and three gun a few years ago. It's a whole new world. What I learned at first was, I'm slow, but I'm accurate. Since then I know work on speed more than anything else. Now it's about shooting fast, and accurate enough. If it's a 6 inch steel plate you are trying to hit, you can be 2 inches off center and still be good. Time to first shot, target transitioning, reloads are all other shooting skill just as important as sight picture and trigger control in the right shooting environment. That said I also see a lot of people at the range that don't have any skill in sight picture or trigger control. They are all skills with different levels of importance for difference styles of shooting.
I'll leave it with some advice from my father. I asked why he preferred using his 1911 in the "other center fire" portions of bullseye matches and not a 9mm or his S&W Model 52 (which was more accurate). He said 2 reasons: 1. he could perfect his shooting with one gun and not have to thick about different weight, feel, sights. And 2. .45 is bigger bullet, I can be further off and still get the X. The point is, "fast" and "accurate" are all relative.