I also watched the theatrical cut of The Big Red One. I've never been interested in the other version.
I'm going to start by springing off of verbALs comment. I've seen 11 by Fuller, recommend 5 and my two favorites - The Big Red One and Pickup on South Street - demonstrate a lighter hand compared to typical Fuller fare. There's a great example of typical Fuller in Big Red One at the insane asylum, one of my least favorite scenes in the film. One of the inmates picks up a machine gun and starts to kill everyone. The point is made just fine, war = insanity. However, Fuller goes the extra mile to have the man shout, "I am one of you. I am sane."
That enforced connection of theme is what makes Fuller so distinct, and it forms the arteries and veins of Shock Corridor. Like Paul Verhoeven, he's not just pushing buttons, he wants you to be aware that he knows he's pushing your button and even though you know what he's doing, it's not going to stop him. It encourages him. This is why I come down on not being a fan of Fuller. He's too happy to rely on cheap and easy effects at the expense of three-dimensional characters. (I'm thinking specifically of the opening scene to The Naked Kiss and all of White Dog.)
The Big Red One isn't nearly as exaggerated, which isn't to say Fuller has lobotomized his style. I love the image of Lee Marvin disguised as an Arab, chomping on his cigar through a layer of wardrobe. Marvin tells his men that "killing is not murder" and in the very next scene we hear a German leader tell his soldiers the same thing. That's Fuller all over. See also the French leader who insists his men should fire on the arriving Americans, including the image of the leader dead, but still firing at Marvin and his men. It's blackly absurdly funny, but just inside believability.
Compare the way authority is treated in that scene to a later one where the team is looking for snipers and Marvin puts one of his best men on point. ("You know how to smoke out a sniper? You send a guy out in the open and you see if he gets shot.") The soldier then puts Marvin in mock danger and that one-upsmanship actually earns the respect of his superior.
The only time Fuller takes it too far is in that insane asylum and during a disorienting hospital scene where somebody comes looking for Marvin and he escapes in a shootout. (I honestly don't know what was going on there.) It's also typical of Fuller that he keeps it small. There are no epic battle scenes, and even the ones that normally suggest a large scale (like D-Day) stay focused on the five core men of the 1st Infantry. It's weird that these five seem to travel across all four corners of Europe. It's a theatrical convenience that I could see someone having a problem with, true as it may have been.
Teproc, if your eyes haven't glazed over you may have gotten that The Big Red One isn't a parody, but Fuller's absurdism does skewer the story in a satirical way. Scenes like the opening kill by Marvin and the way it comes back at the end with Marvin threatening, "You're gonna live if I have to blow your brains out." I don't think Fuller has made an anti-war film, but one that repeats over and over that the very idea of war is insane. That you can justify killing as being different from murder, that it's your job, but that's something only an insane person would do.
Your review smacks of things I would likely say if this was one of my first by Fuller. The only place where I specifically disagree with you is that concentration camp finale. I thought the images, with the soldiers surrounded by sunshine looking at the lean, haunted, ghost-like faces surrounded by shadow and ash clouds, is Fuller at his best. Those moments really grounded the film hard, right when it needed to. It humanizes Lee Marvin's character in a way you don't expect Fuller to even try. You can (and do) argue that it's wrong to have the boy there to suit the needs of the American soldier, but that's who we've been following. It's not the kids story. That's a different film.