I don't know why I like to set myself on fire like this, but here goes. I think this whole movement thing has gotten way outta hand. Let's look at the "Big 3", no actually let's look at 5 moves.
Presses, there is no way anyone is going to get the same benefits out of benching as they are out of the military press. And if I were only going to do one, I'd do dips. But then look at the snatch, I guess they consider it a pull but you can't tell me you're not going to get some of the same benefits out of snatching that you get out of military presses.
Pulls, how many different pull exercises are there? You don't get the same benefit out of low rows as you do lat pulldowns. Heck you don't even get the same benefit out all the different pull up variations that one can do. It definitely depends on your needs as to whether one would or should do deadlifts as compared to cleans.
Hip hinge, is the deadlift a hip hinge or a pull? Swings - hip hinge, snatch - hip hinge but what about the shoulder work it provides, bent over rows - definitely a pull but what about the stabilizing muscles of the posterior chain.
Squat, I don't think anything makes up for not squatting. I don't care how big your deadlift is, the only way to get a big squat is to squat. But what kind of squat? Now we're somewhere, goblet, front, back? You are going to be squatting with weight in front of you much more than on your back. But are you not using your posterior chain to back squat?
Carries, if you can press your weight and deadlift 2x your weight I'm guessing that you can carry stuff pretty well for your needs. Otherwise then it's a whole conditioning thing and carries would only be a part of an all around general conditioning program. And to limit carries with your hands rather than your arms the the weak link is your grip.
So we go way back in time and listen to what the old timers had to say. If you just want to get big, squat, bench, and bent over rows, deadlift from time to time. What if I just want to get strong? Strong at what, nail bending? Better work on those hands young man. Yea, but I want to be in good condition too. Good condition to do what, carry a sixty pound load 10 miles? Better start on carrying loads for distance and get some good shoes. It just all depends. But wait look there, there's a man on fire.