Taking Defensive Handgun II at TR in Texas, we were drawing and firing short strings, usually 2 shots, COM, at about 5 yards. My shot groups were 4-6" and that was far too large for the instructors. My assigned partner was doing groups of 2-3 or maybe 2-4" as were many, but they were often shooting AFTER I had finished my string. That isn't to say that I was fast. My draw was actually slower, but my shooting faster. Once my sights initially settled COM, I would double tap. What the other shooters were doing was more of letting their sights settle, then fine tune correct for minscule error, then slow fire one shot, then two, or double tapping, but not until the first shot was perfect.
As this was all going on, the instructors on the PA were barking the Earp misquote of "Fast is fine, but accuracy is final." The misquote is right, but completely ignores an important caveat. That is, accuracy doesn't mean crap if you are too dead to be accurate. I may not have been a very good shot compared to my fellow students, but if I was fighting them in a gun fight, landing 2 shots on the my opponent, COM and within 6" of one another will minimally be messing with their ability to aim their perfectly aligned shots and very well may land them as completely dead before they have a chance to shoot.
No doubt smooth is fast. The difference in my shots (and 2 or 3 others) and those of many of the other students was in the range of a quarter or half second between when my shots started and theirs as the majority of the class would just about shoot together in a resounding thunder. That amount of time is what it takes many folks to let their sights fully settle.
The critical aspect here is shooting good enough to handle the job. Nobody measures distances between impact and X rings. So at 5 yards, for example, do you want to spend much extra time getting that perfect 2-3" group knowing that you are close enough to the bad guy that he may very well be able to start landing shots on you as a matter of proximity and spray and pray tactics? How well are you going to be finishing up the sight alignment when the first impact on you occur?
Ken Hackathorn had us do a nifty test at 10 yards. He had us aim perfectly, then move the gun so that the front sight moved fully left (but still aligned vertically) until we could just see the edge of it in the rear sight notch, then fire. We then did it to the right. Next we lowered it all the way until just barely seeing the top, but had it centered horizontally, then we fired. Last, we raised the front sight until the base of the front sight left the rear sight notch sight picture and then fired. All 4 shots landed on the silhouette at 10 yards. For most guns with proper sights, if you can see any part of the front sight in the rear sight notch and actually intend COM, you can hit the opposition at 10 yards. It may not be a good hit, but you can hit. On my gun, the impact area, using the corners was roughly 20" tall by 20" wide. That is a 400 square in area.
So if you have that much slop at 10 yards and can hit, at 5 yards you will have potential max impact area only about 1/4 the size of the impact area at 10 yards (reducing both vert. and hor. dimensions by 1/2 for the reduced range, assuming your sight notch is square). So the box is now 10x10" as max size for seeing the sights in the notch, or 1/4 the area.
Now, at 5 yards, if you can see half of your front sight in any of the 4 directions, only half, you have a 5x5" potential impact area or another 1/4 of the previous size. So that was about what I was doing, sometimes better, sometimes worse, but close. Now, take a 5x5" square and put it on your person COM. Would you be happy landing 2 shots in the box on a bad guy if you could do it BEFORE he shot you, or would you rather wait until you can line up the idea shot...keeping in mind you don't know the skill of the bad guy, but you know he is more than close enough to be effective via luck.
Hackathorn used this as a training point to help us shoot better, but it also showed just how far away a person could be and have crappy of a sight picture that person could have and still shoot you, maybe not lethally the first time.
Funny thing, the same gun school gurus really tight shot groups in practice will tell you that you will speed up in real life. They also tell you that you will shoot like you have trained. Okay, so you speed up. Given the circumstances of the fight, your opposition has undoubtedly sped up as well. If he speeds up as much as you, you may still be wasting time working the perfect shot since your increased speed may not be any more than his increased speed.
Of course you have to hit well enough to be effective, but spending time for the really precise "accuracy is final" shot may be what end up being final for you.