I would say that the VAST majority of casual shooters have no idea how accurate their pistols really are. When I see people practicing at 7 yards with a timed fire pace and missing the silhouette and then blaming the gun, it is incredibly silly. You'd have to have the barrel flopping around in the frame for it to be the gun.
Certainly a machine rest 10-shot group at 50yds the size of the X-ring is something that you should expect to pay a little extra for with either a custom pistol or some aftermarket work. I would guess that most off-the-shelf pistols should easily keep 10 rounds in, say, 6 inches at 50 yards from a machine rest, and that would translate into about 1 inch at 7 yards. Anything that falls outside of a 1 or 2 inch group at that range is a flinch, poor trigger control, or poor sight alignment.
When I hear people say that they have a "really accurate" pistol because they are keeping their 7 yard groups around 3 inches or so makes me think that they have found a pistol that is easier for them to shoot accurately than their other guns, since the pistol itself is likely being shot nowhere near it's capability.
Certainly some of the top shelf pistols are not only more accurate, but easier to shoot more accurately with great sights and triggers. I have a 1911 that I can't seem to tame because the trigger is heavy and gritty enough that my body figures out how to flinch within about 10-15 rounds and everything falls low and left by several inches at 25 yards and that is WAY more than any mechanical problem with the gun. It's all me. Regardless, I bet I will shoot better with it after I have the trigger redone, even though the pistol itself won't be any more "accurate" than before the trigger job.
I'm a poor bullseye shot, and the best group I've ever had was 100 7-X on timed fire at 25 yards during a match with my .22 High Standard. 10 ring is 3.36 inches. This was 1-hand, standing, outdoors. At least half the guys in my bullseye league shoot better than I do, and I don't find any of those targets above to be extraordinary from a skilled pistol shot. Certainly they came from a good day of shooting.
Oh, and to answer the original poster's question, I like to shoot at 50 yards slow fire both because it forces proper slow fire technique and because I'm practicing for Bullseye shooting. If I want to work on self-defense stuff, I change to a silhouette target at 7 yards, but I really don't find that sort of shooting to be as much fun so I don't do it much unless I have a tactical class coming up.
-J.