I think it's a little more basic than what is proposed. Let's go back to the actual lines:
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KIRK: Gill. Gill, why did you abandon your mission? Why did you interfere with this culture?
GILL: Planet fragmented. Divided. Took lesson from Earth history.
KIRK: But why Nazi Germany? You studied history. You knew what the Nazis were.
GILL: Most efficient state Earth ever knew.
SPOCK: Quite true, Captain. That tiny country, beaten, bankrupt, defeated, rose in a few years to stand only one step away from global domination.
KIRK: But it was brutal, perverted, had to be destroyed at a terrible cost. Why that example?
SPOCK: Perhaps Gill felt that such a state, run benignly, could accomplish its efficiency without sadism.
KIRK: Why, Gill? Why?
GILL: Worked. At first it worked. Then Melakon began take over. Used the. Gave me the drug.
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So, "efficient" was not about the government... it was about the organization and rallying of the Nazis that bonded a nation to work together towards a common goal. Unfortunately, their ultimate aims were not benevolent. They invaded benign neighboring countries. They turned sadistic and murdered millions.
The idea of the episode story is intriguing, but it is flawed... it does not take into account the unique series of circumstances on Earth at the time. Certainly on this alien planet, you wouldn't have the exact same circumstances. There's little guarantee that introducing the Nazi movement would accomplish Gill's objective.
Remember at the time that the United States had a comparatively piss-poor military. Our first venture into the war was astonishingly 3rd rate. But we had a propaganda machine of our own that rallied the citizens. It was highly honorable to participate in the war effort and give your all. And our nation rose up quickly. It's really impressive when you look at how fast our military grew in skill, resources, and technology.
So, the efficiency was really all about unifying the country towards a common cause. The Nazis were on a path of "taking back what is rightfully ours" and Americans were united on "ridding the world of fascist tyranny". For the American version, there would need to be a facist regime to be vanquished, which there wasn't. For the Nazi version, there needed to be a chaotic nation needing to be united against an oppressor--which there was. So, the Ekosians were a primitive, warlike people in a state of anarchy. The other planet, Zeon, had a relatively high technology, and its people were peaceful but suppressing the Ekosians. Gill sought to rally the Ekosians beyond their anarchy and rise to the capable level of the Zeons to balance things out and achieve peace... but the Ekosians grew too strong for the Zeons and turned the tide. We never got to really understand the nature of the oppression and what Gill conceived in rallying to unite the Ekosians... was it Melakon that invoked the racial purity concept and make the Zeons out to be the inferior race that needed to be wiped out? Unfortunately, the episode leaves many unanswered questions.