I hear and read this ^ constantly, and I agree with it in principle...except...
...I really think you need to clarify what you mean by "shoot as good". How many shots are we talking about?
I'm also an admirer of the .375H&H, the .338WM and a bunch of others that are spoken of in hushed whispers by many posters. If I were to sit down at the bench and start shooting and shooting and shooting, there is absolutely no question that I would reach the point where my accuracy started to go to hell sooner with a .375 than with a .223; I wouldn't dream of denying that. I also won't deny that a drop in accuracy will eventually occur with the .22 as well...but it's gonna take hundreds of rounds at a sitting, so will likely never be seen.
But...and it's a big "but"...if you have practiced with a bigger gun and become accustomed to firing it, there is absolutely no practical reason why you should not be able to fire a group or a couple groups or a box of ammo with it and be every bit as accurate as you are with a .22 rimfire. None. Zero. Note that I said "practical" reason.
But...there's still the
imaginary reason. If you pick up a .375 and shoot a 6-inch group, it's because all the doomsayers have messed with your head; you "know" it's gonna hurt, because they've all told you it will; some of them "know" it will because they've actually shot one...poorly...but most of them are just passing on this wisdom from other terrified experts. By the time you load up and aim and finally squeeze the trigger, you're breathing hard, your anal sphincter is clenched like a fist and you're sweating bullets. Of course you're not going to shoot it well.
I know I can shoot a cold group from my favourite .375 that is every bit as good as that from my .223 coyote gun; several groups, actually. I know that I can probably get to about 15-18 shots before I can see and feel my concentration and accuracy starting to go downhill with the bigger gun. So guess what? I stop before I get there! I'm never going to be doing high-volume shooting with the .375; it's not a ground squirrel gun, it's a big game rifle and as long as I know what my limits are it's easy to stay within them and to enjoy shooting it. For the one or the couple of shots I will take on a hunt, I will be thinking about the game and my sight picture and my hold and breathing; I won't be quaking in fear because, oh my God, I have to shoot this horrible rifle again! I can concentrate on the shot.
One of the pea-shooter guys will likely make some sarcastic crack that it's actually my .223 groups that are just as bad as my .375 groups; that's okay, believe what you want to believe. Just remember that belief and truth are not always, or even usually, the same thing. Practice, learn to shoot the bigger gun, don't overdo it at any one time to the point where you start to finch, and you will find yourself shooting it very well and, yes, enjoying its use on game. And at that point, trust me, you won't give a crap about what the naysayers think.