What was the significance of Kurtz joining airborne

I think the key is the line "He could've gone for general, but he went for himself instead."

Kurtz had apparently lost interest in climbing higher in the military ranks, which presumably would have taken him away from combat and the front lines. He seemed to want to go in the opposite direction -- toward the heat of the fight -- so Kurtz requested Airborne training as a first step toward returning to combat.

Kurtz took himself out of the running for promotion by enrolling in Airborne training. Promotion would have required continuing to follow an advancing career path -- whereas Kurtz brought his career advancement to a halt by enrolling in Airborne training.

Being qualified in a airborne course expands your application as an active soldier. That's what Kurts wanted, be active, embrace the war, the weight of anything that would be necessary to do in order to win the war.

After he joined the special forces, with the training in how to mix with the local population, convincing them to fight against the NVA and VC under your advice, he gone to the final part of his process of taking the war by the hair and winning it. The joint operation he conducted shows to the Higher ranks that He meant business, and whatever he needed to do in order to finish his business.

Being a airborne gave him the mental strenght to become the leader he needed to be to win the war. After it, he , in my opinion, was only about gathering the techniques on how to make the vietnamese fight for what he believed.

It wasn't Airborne that killed his promotability, it was Special Forces.

The first pertinent quote (from the script) is:

WILLARD (V.O.): At first, I thought they handed me the wrong dossier. I couldn't believe they wanted this man dead. Third-generation West Point, top of his class...Korea, Airborne, about a thousand decorations, etc., etc. I'd head his voice on the tape and it really put the hook in me, but I couldn't connect up that voice with this man. Like they said, he had an impressive career. Maybe too impressive. I mean, perfect. He was being groomed for one of the top slots in the corporation. General, chief of staff, anything. In 1964, he returned from a tour with Advisory Command in Vietnam, and things started to slip. His report to the Joint Chiefs of Staff and Lyndon Johnson was restricted. Seems they didn't dig what he had to tell them. During the next few months, he made three requests...for transfer to Airborne training, Fort Benning, Georgia...and was finally accepted. Airborne? He was thirty-eight years old. Why the fuck would he do that? 1966...joins Special Forces, returns to Vietnam.

The second pertinent quote is:

WILLARD (V.O.): Thirty-eight fucking years old. If you joined the green berets, there was no way you'd ever get above colonel. Kurtz knew what he was giving up. The more I read and began to understand, the more I admired him. His family and friends couldn't understand it. And they couldn't talk him out of it. He had to apply three times, and he put up with a ton of shit, but when he threatened to resign, they gave it to him. The next youngest guy in his class was half his age. They must've thought he was some far-out man humping it over the course. I did it when I was nineteen, it damn near wasted me. A tough motherfucker. He finished it. He could've gone for general, but he went for himself instead.

Now, for the why any of that is important. You have to look into the history of the US Army a bit. Airborne Infantry was a thing newly developed in World War II. While there were smaller applications earlier, the first major Airborne drops were during D-Day, inland from Normandy. They went, by and large, fairly well. Then during Operation Market Garden, there were more Airborne troops dropped, which went, by and large, again really well. So Airborne and the philosophy and change to Army tactics that were developed from it, were well accepted. By the years before Vietnam, the only officers who would get ('full bird') Colonel were those who had the political backing to continue on to Brigadier General. A Colonel in those years would have served in Korea, and an Airborne Colonel would have been viewed as the more well rounded officer. Not having Airborne wouldn't have precluded promotion, but Airborne was the "Up and Coming Young Officer's Thing-To-Do."

Now let's discuss Special Forces. Like Airborne in the beginnings of WWII, during the years just before Vietnam, Special Forces was only just then being built up. (Kennedy did so, for many reasons, not the least of which was the glamor effect). The origins of Special Forces were OSS, and various other special groups, usually called Commandos. The problem with the OSS and Commando groups were that the regular senior officers didn't have a clue about what to do with them, or how to use them. (A good movie to watch for this specific thought it The Devil's Brigade. If you watch it, pay close attention to Carol O'Conner's character and how he treats the Brigade.) Those troops and their officers were Prima Donna's. They never really had a chance to shine enough beyond single missions to evolve a philosophical change in Army tactics.

So, now we have Kurtz. He was a golden child. Being groomed for Generalship. On the fast track. Airborne was a way to help achieve that. There was no problem at all with his Airborne training and wings. In fact, they would have made him even more likely to make promotion.

Special Forces? Not so much. In fact, with that, he instantly became un-promotable. There simply were NO generals in or from the Special Forces branch at all.

Finally, when you read the line

Airborne? He was thirty-eight years old. Why the fuck would he do that?
That was simply because Airborne training was, and is, very extremely physical and a thirty-eight year old is a grandfather compared to all of the 17-18 year olds.

(Of course, by today's standards, a Special Forces Colonel has a better chance at promotion than not, because today, with the current unconventional warfare doctrine, it's not the major Army commands fighting the wars as much as the SF groups. But that's just my opinion as a retired Army NCO.)

Sedition. When Julius Caesar went to Gaul he did so with the sole purpose of him becoming emperor of Rome. And his men backed him up. Listen to the respect, and awe that Willard has of Kurtz. Kurtz is dangerous to military order. Kurtz is dangerous to the chain of command.
"he could have went for general" - he could have maintained military order. "he went for himself instead" - general isn't good enough for him. He has got to be more. Like Caesar who couldn't just be consul or a great general. He wanted a kingdom. its seditious.