John Smith would have never ascended to such a high rank

On the first point, John Smith was fighting for the allies before the bomb was dropped on DC. Loyalty changed from U.S. to Nazi. Clear as day.

On the second, while yes, Göring was an adult by the time he joined the NSDAP.... well, not so much "joined" as he was one of it's founders at Hitler's side from the very beginning. And, as a founder, combined with the fact that he was a defeated combat pilot in WWI and needed to restore his national pride; of course he'd be fully invested in Nazi ideology. The fact that he fell out of Hitlers favor is irrelevant to my point. Unlike, John Smith who had to embrace it as a matter of defeat and survival. Comparing Smith with Göring is simply an exercise in false equivalency.

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Yes, it would have been very difficult to administer the Nazi regime in America using only native-born Germans in all the positions of significant power. Moreover, such an arrangement would have fostered the alienation of the local populace, making governing more difficult. Americans coming to maturity after the end of the war would still have been too young to run the country in 1962, so utilizing men of Smith's generation would have been essential. And some guys are just inclined to follow orders in any environment, as long as they themselves and their families are secure.

I think that striped band depicted his higher rank, instead of lower birth.

I think the arm band with red stripes is because he is part of the American-SS. German born SS (thus part of the German-SS regiments) working in the States (e.g. look at the SS working in the Nazi embassy in the Pacific States and have German accents) wear the all red armband in this reality.

In our reality foreign (non german) SS regiments had designations on their lapels and sleeve cuffbands. However this was during WWII when the SS were increasingly seldom seen wearing the black uniform and instead opted for the SS grey service uniforms, therefore slightly different rules as SS didn't wear armbands with the grey service uniform.

https://en.wikipedia.org/wiki/Uniforms_and_insignia_of_the_Schutzstaffel

Having read the above thoughtful replies, I have reconsidered my initial assertion and can now find plausibility in the situation.

A few points that weighed heavy in my thinking:

-It's made very clear that John's primary drive to every waking moment is his family, and after seeing the DC detonation from the cabin where his wife and new baby looked on, it's very likely that in that moment he made the decision to switch allegiances as a matter of his family's survival and after making that decision would be fully committed.

-In Season 1, he had a conversation with Rudolph Wegener in which they lamented about the "things" they had to do during and after the war in order to further the Reich's vision. It was very much implied that Smith partook in the atrocities, and in doing so cemented his status with high command. His role in mass murder was also corroborated by Resistance, specifically Dixon when coercing Jules to cooperate.

-As dvd123 pointed out, 15 yrs is a decent amount of time to show commitment to the cause, and as others pointed out, having an American very visibly furthering the Nazi agenda would aid in winning the hearts and minds of the population.

-I'm sure in the wake of the Nazi victory and for some time after, Smith was under heavy surveillance and scrutinized, while him being very intelligent and Machiavellian, he would have known this, and strategically committed said atrocities, among other things, in order to gradually have his loyalty less questioned, until it wasn't.

To be sure, he's an evil self serving Nazi S.O.B., but I think the writers and Sewell himself did a great job in creating a semblance of empathy for him. He seems less about the Nazi ideology and more about the safety and success of his family (which we can all relate to). With the softening and wisdom of age, recent experiences with vulnerability (Thomas's condition), witnessing acts of mercy (through Juliana), resulting in ideological contradiction and ambiguity, all lead his conscious weighing heavy enough to intervene to prevent a nuclear holocaust.

Or......

He knew his and his family's future would be in peril under a Heusmann regime as his opposition to Heydrich was surely well known. So, consistent with the pattern of his behavior, he preemptively struck.

Thoughts?

That was a lot more true during WWI than WWII. Yes, there were pockets of German culture at the time. Still are/were as of recently. In Texas where I live there are areas just a few hours from Houston where the radio stations were in German (or Czech) until a couple of decades ago. But that was a rare thing. What kept America out of the war in the 1930s-40s was not pro-German sentiment, but isolationism.