There are two different, very distinct, issues here. The first is, of course, Yamamoto being in actually command of the Kido Butai, with the second, the possibility of a third strike on Pearl Harbor, being an entirely separate matter.
One of the major misconceptions that surround Yamamoto is his "gambler" mentality. This actually is not a proper image of the man. He was a man of considerable earthly pleasures, when off duty, but a reckless gambler he was not. He was a poker player, which, if done well, in not so much gambling as evaluation of the other players. The Admiral was a very good poker player, one trait of successful professional card players (and Yamamoto, who supported a lifestyle far beyond his pay grade with poker winning, certainly qualifies a a pro) is that they actually never gamble; they play percentages. The percentages at Pearl Harbor had already been exceeded, the Kido Butai had suffered very light losses in personnel (although roughly 1 in 4 aircraft available before the battle began had been rendered Hors de Combat (29 actually shot down and around 75 have suffered significant battle damage, most in the second wave). The operational requirements for the mission were fully achieved, surprise was well & truly lost, and there were three American carriers unaccounted for (including Enterprise which was close enough that a flight of her SBD's actually arrived during the attacks, resulting in several being shot down blue-on-blue), it was a good time to cash in and leave the table. While, in hindsight, Yamamoto stated that Nagumo had missed an opportunity, at the time he supported Nagumo's choice. Every good card player knows when to call it a night.
Yamamoto MIGHT have pushed a third strike, but his actions would likely have been the same as Nagumo's. While certainly a planner of some vision, and even more a good listener to his staff, Yamamoto never exercised direct command of the Kido Butai during the war. His actions at Midway, where he initially ordered that the invasion proceed according to the original Plan, even after all four fleet carriers had been sunk; not instructing the amphibious forces to wait for the Combined Fleet to reach direct support positions, or even detaching the heavy escorts that had been shepherding the now lost carriers to support CruDiv 7 in their bombardment tasks shows clearly that he was not incapable of making a poor decision (truthfully, the entire Midway Operation is indicative of this). Moreover, as will be discussed, the decision to retire was NOT as poor as some believe.
The American base, while seriously damaged, was not entirely supine, not even close. While the fighter defenses had been seriously degraded, there were still better than 40 fighters (including 27 P-40) fully operational. The Army AAA batteries, all 31 of them were manned and armed (in the initial waves, only four of these batteries had been able to get into action), while the remaining American vessels in harbor were either fully manned and ready for trouble, or they were about to sortie. In the second wave, the Japanese had lost 20 aircraft and had suffered an additional 60 or so mission kills, this was while the confusion from the initial attack was still at full bellow. The Pearl Harbor defenses would have had several HOURS to have gotten back (or initially) into action, since the Japanese would have had to refuel and rearm the entire strike package before it could be launched. With the edginess of the AAA crews (who shot down a number of AMERICAN aircraft) the fully alerted nature of the surviving fighters (an interesting potential fight would have been the obsolete P-26 vs the equally obsolescent Val & Kate) the third wave would have had a very tough sled.
Had the third wave lost 40 or so aircraft, along with around 80 mission kills (or 120 airframes, just slightly worse than had happened in the second wave), it would have knocked at least one carrier, probably two, out of the war for months, as can been seen in how long it took Zuikaku to reconstitute her wing after Coral Sea. The operational loss of one or two decks could have had some interesting byproducts in the war's opening weeks, starting at Wake, and continuing into the Indian Ocean operations of April 1942.
The Japanese Navy only began to see it as a lost chance when the reversals of May & June of 1942 occurred, until then Pearl Harbor had been seen as an unqualified success (save the poor luck of missing the carriers). Moreover, in the long run, even the total devastation of Pearl Harbor would have altered the end of the Pacific War not a whit.
BTW: Enterprise was on the way back from delivering aircraft to Wake, while Lexington was around 400 miles from Pearl delivering aircraft to Midway.
