I'm a little late to the game (also new here, hi!) so I will just try to hit a couple points. In the various reading I have done, a big part of having everyone spread out for Yamamoto was his thinking that if he brought the entire fleet en masse, the USN would stay home. That goes back to another point a poster made (several pages ago) about USN being scared.
As far as doctrine, most of the salient points have been covered. One thing that has always surprised me is why the Japanese never employed carriers like Shoho and Zuiho as solely CAP carriers. They could carry 30 A6M's with little issue and allow the big carriers to focus on what they focused on anyway - offense. I never really understood what good it did for Zuiho to run around with a dozen B5N's or D3A's. They learned something from Midway, because by October 1942 Zuiho was only carrying 6 B5Ns and 18 A6M's...why not take the plunge and just go all fighters and handle the CAP? [as an aside, I have read that not every A6M was even equipped with a radio, but they definitely needed more and better CAP organization and communication]
Why did Nagumo turn toward the USN carriers? Not even just Hiryu, but the ENTIRE group of carriers, once the USN carriers were spotted? When they were recovering the Midway strike and obviously vulnerable? I understand the attack mantra, but that doesn't mean you have to be foolhardy....Japanese carrier planes, even in 1944 (philippine sea) had a significant range advantage over the Americans. Why not turn AWAY, recover the Midway strike, put some distance between you and essentially, you control the engagement due to your range advantage. Yes, I know, you have to turn into the wind to recover or launch, but I am pretty sure the prevalent wind direction was not such that they had to head toward the USN to land the Midway strike. The goal of the plan should have been clear - we're here to sink the USN carriers above ALL else. Midway is the bait. Of course, they didn't think the USN carriers would be there when they first hit Midway.
Someone else mentioned the Aleutians operation. Let's remember that a mere 6 weeks before Midway, the Doolittle raid had hit Japan. They got as close to Japan as they did roughly by using the same tactic the Pearl Harbor strike force did - stay north out of the sea lanes and more out of sight. Kiska, Attu etc were seen as another extreme route to do the same thing, and bases to possibly help spot such a force. As far as a distraction, that's counter to the decisive battle plan, isn't it? The Japanese WANTED the USN carriers to come to Midway, they wanted to destroy them. I have read an account or two that stated that hitting Dutch Harbor etc was intended so that the USN carriers engaged at Midway could not interfere...but I have a problem with that cause they hit Dutch Harbor FIRST, on the 3rd of June. In any event, it was not an operation designed as a diversion.
Finally, the point of the post, why did Nagumo keep his job? A good question, cause Inoue lost his after Coral Sea, which was certainly more successfully than Midway was. Similarly, even though he smoked the allies at Savo Island, Mikawa was eventually rebuked for not hitting the transports and eventually was blamed for the slipshod Guadalcanal campaign and relegated to obscurity largely. So who's left to command Japan's carriers after Midway? Other than Ozawa, I think Nagumo was the only vice admiral with carrier experience. I think the Japanese system could not put a rear admiral in charge of the carrier forces with an experienced vice admiral around. Many have made the point that given the conflicting orders given for the Midway operation, there was little any commander could do to achieve victory, but Nagumo had proven to be conservative and somewhat timid since Pearl Harbor, but he was given a LOT more chances, it seems, than any other admiral in charge of carriers. But if it was me, if I was Yamamoto, after Midway, Ozawa gets the job in July, not November