I'm not a longrange hunter, but I am an admirer of bigger, heavier bullets in general. Won't say "magnum" because that's just a marketing term that doesn't define anything precisely, but as an example here I'll say that the .300WM is likely my favourite cartridge for most hunting use; in my case that means deer/moose.
I keep hearing it repeated that "most" people...whatever that means...will shoot "better" with a 6.5 as opposed to a .300...and, by extension, that a .338 will be even worse. That's obviously true when we're talking about extended shooting sessions; the big gun will be more fatiguing more quickly, and the results will show that. How soon those results become apparent is a function of practice and experience, but it's going to happen.
But that's not hunting. Hunting typically, ideally, involves just one perfect shot; one or two more might occasionally be called for, but that's pretty much it 99+% of the time. And the simple fact is that a single shot in a hunting scenario, or a couple shots, can be fired just as accurately with a .375H&H as with a 6.5Whatever. If the single or couple shots turns into a string of shots, then absolutely, the big gun will make it more difficult to maintain that accuracy than the smaller one...but that's not the goal anyway. We want the animal down and dead as quickly as possible
Many people who don't or won't own the big guns like to mock those who do; it often sounds like a case of "if I can't do it, there's no way that you can do it!" A lot of those folks have never even tried it, but they "know" that it can't be done. I think a big part of the problem is that many shooters can only shoot at a rifle range that requires driving a fair distance, and so when they go they want to pack as much shooting as possible into the range time they have. They aren't willing to only shoot a couple groups out of a particular rifle and then put it aside...but a bigger gun might need to be treated that way. Thus, they exceed their own personal limits for that particular gun; the cumulative effect of recoil destroys their concentration and focus, their accuracy degrades, and the immediate reaction is that "I can shoot the little gun much better!" Of course you can; if you want or need to shoot group after group after group, the little gun makes it easier for you to do that...but it doesn't mean the big one can't be shot just accurately for a shorter time. It just means you can't shoot it as much on any single occasion as the small one.
Those of us fortunate enough to be able to shoot close to home, or at home, can space out our shooting, keeping each session short enough that fatigue doesn't become an issue. I shoot virtually every day, although the bulk of it is with rimfires. When I was preparing for an African hunt, I shot a bit literally every day with my .375 for several months before hand. Over time, the number of shots I could shoot effectively increased gradually; by the time the trip arrived, I could comfortably shoot 40 or 50 shots before my mind started to wander and my accuracy went downhill. But...that was more shots than I fired the entire time I was in Africa, including the initial rifle-check upon arrival, so it was more than enough.
I'm not suggesting that the smaller guns aren't effective; in most cases they will do the job, but there will inevitably be occasions when the big guns are superior as long as you can shoot them effectively. Whether you want to use them because a bigger, heavier bullet is superior...or because the nostalgia and tradition of a cartridge like the H&H makes a hunt more enjoyable...or because a specific rifle you like happens to be chambered in one of the big boomers...well, these are all valid reasons for many of us...whether many of you believe it or not.
Everybody should shoot what they want, without insisting that everybody else is wrong.