How did Smith become a high ranking Nazi?

How doesn't it make sense? They live in what is essentially a perfectly functioning country in many ways better than the real life counterpart. After decades of propaganda and brainwashing, combined with the fear of authority, only the small minority will remain rebelious, and that is true virtually everywhere. There were plenty of nazi supporters in america before the war and without the daily propaganda pushing american patriotism, patriots kinda die out.

I'd also take the history of Yugoslavia as a good example - at the beggining of ww2 nazis marched into the Croatian capital and they were greeted with roses and flowers. Fast forward 20 years later (cca 1960s) everyone is an ardent communist. And all of that despite the fact that American Reich looks like an utopia compared to this era of Yugoslavia. People hang propaganda posters on the walls of their homes. In the 80s, when the great leader Tito dies, every woman in the country cries her eyes out. Then we have the 90s, the fall of communism under external influences and guess what - after less than 5 years there's suddenly not a single communist in a country. Even people who were high ranking officials in the communist party are now pretending to be hardcore church going nationalists.
Lesson? People follow the authority and the type of people to become high ranking officials care about their careers first and foremost, not the ideaology.

Actually they showed what happened with Smith. He was not a kid. He was a high ranking American officer; and the conquering Germans were smart, they offered everyone a choice: a good position in the new order for those who pledge loyalty to the Führer. Yeah I'm sure there were problem cases at first. You don't think they wouldn't keep an eye on these people until they were sure who was truly on the team, do you? I'm sure they probably arranged for most of them to see a chance to betray their oath and do some real damage - anyone who jumped at it, of course, got a bullet in the head.

Having an occupied territory run mostly by people who are from that place instead of the outside is good policy. While they may be viewed as collaborators by the diehard rebel types, overall that policy greatly reduces the inevitable resentment and tension. A few decades later you can see with Smith's children what the result is. Two of the three were thoroughly indoctrinated. Keeping a foreign population under the gun all the time, every day, is less effective than giving them hope that they can achieve success under your dominion as long as they follow the rules.

You have a family to support. You have career ambitions. Do you throw your support to a lost cause and probably get all of them killed, or join the Nazi Party, sieg heil along with everyone else, and have the best life you can under the circumstances? Human beings have always become loyal supporters of those in power, while they're in power. Regardless of their personal feelings. It's how they stayed alive and made a living through even the worst historical periods.

The premise is fine. Smith was a good soldier and a leader of men. He just sold his soul to the Nazis to save his family. He committed atrocities (hinted at in the script) to prove his loyalty, but didn't believe the Nazi propaganda. It was a career move to him. Maybe he would have tried to make things a bit better for Americans when he was in power, compared to what pure Nazis would have. At least the show teased you with that thought, or let him delude himself into thinking that. His subordinate and friend played along too, but turned on Germany as soon as the odds were in his favor.

He kissed ass, did what he was told (and then some) and eventually, in the eyes of the occupiers, became "one of the good ones".

The book did not do that, or anything like it. The book had its own problems. The series was bad, except when you flash around alarming symbols in a TV show it gets a lot of mindless attention while people try to find a way that it makes sense.