It has been said that the idea of the fast draw in the old west is a myth. Gunfighters didn't "do the fast draw."
Well, I can certainly believe that to the point that the image of, say, Matt Dillon approaching somebody on the street and at a set time good guy "lets" bad guy go for his gun first (after giving fair warning) and then good guy still manages to outdraw bad guy because good guy is just that good. Nobody, good or bad, would approach a situation in that way and expect to live long.
It is said the closest thing to that happening was not a "fast draw" situation but rather a face off or shoot off between Wild Bill Hickok and a Confederate. Hickok managed to place a .36 caliber bullet in the Confederate's heart at something like 50 yards away from his 1851 Navy pistol.
HOWEVER, all that being said, a true gunfighter or lawman would definitely need to be "fast on the draw" or (more to the point) quick with his abilities just as anybody today carrying a handgun would want to be proficient with their weapon. Modern gunfighters stress this such as Bill Jordan, Jelly Bryce and Ed McGivern. Men of the Old West knew were aware of this too. If you realize you're in a bad situation, you need to get that gun out as quick as you can to respond to the threat (not wait for the bad guy to draw either).