Distance is just an input into all the other factors you weigh. Your eyes are the most important, but when your eyes and the yardage disagree, you have to think about what might be fooling your eyes, or if it is possible the yardage is inaccurate (the latter is probably not an issue for the pros, but it can be occasionally for us amateurs relying on sprinkler heads or yardage books)
If you are playing a shot with a lot of run, a lot of wind, a lot of elevation change, etc. the yardage becomes less important, but I don't think it is ever truly useless to know it. If I'm in the trees and hitting a low hooking punch 2 iron, I want to know if I'm 130 or 150, just knowing the distance influences me to hit it harder. I'm not thinking "oh 150, I need a 60% punch instead of 50%" but its in my head as I plan the shot and as I address the ball. Its not nearly as important as knowing the difference between 130 and 150 when I'm in the fairway and hitting to a soft green on a windless day, but it still matters because the shot that would work perfectly from 130 in the trees may not even reach the green from 150.
I actually like to know how far I am on flop shots around the green. Assuming I have time, I'll walk up to the hole and figure out where on the green I want the ball to land. The hole is right there, I can see it just fine, but as I take my practice swings and hit the shot I'm thinking "OK, carry the ball 16 yards". I don't know that this really helps, since I'm not sure I really do better on those shots than on those where I just walk up to the ball and hit it (obviously that's more likely on shots I've left short of the green) I do know that occasionally I'll concentrate too much on the flag on a flop shot instead of thinking about where I want the ball to land -- with the result that I usually carry it just short or slightly beyond the flag, which isn't always going to produce the desired result!
I figure every little bit of information helps, so long as you don't overrate its importance.