The main character is a human who has risen to be the assistant to the alien in charge of North America because he can understand their inscrutable thinking a little better than most. It's a mystery why the aliens have taken over Earth as they don't seem to want anything from Earth or enjoy being in charge.
The story you are looking for could be All the King's Men by Barrington Bayley. It appeared in a collection of his short stories published in 1978 called The Knights of the Limits. Someone has put a reading of the story on YouTube here. It can also be read here, along with the rest of the stories in The Knights of the Limits.
All the King's Men is different from what you describe in one or two respects. The main character is indeed a human who has risen to be the assistant of an inscrutable alien in charge of a human country, but the country is Britain, not the US. Furthermore it is not the whole Earth that has been taken over but only an apparently random selection of countries: Britain, Brazil and South Africa. The first two were conquered by different and mutually hostile groups belonging one alien species, the third simultaneously taken over by another species entirely.
However the incomprehensibility to human eyes of why only three countries should be taken over does tie into what you said about the inscrutability of the aliens and the mystery of how they don't seem to want anything from their conquests. The main theme of the story is the gulf between alien motivations and human ones.
The main character and narrator is known only as Smith. He is now the chief human lieutenant to a seven foot humanoid but vaguely insectile alien known as the King of All Britain. Smith is conscious of how inadequate he is to fill this role compared to the recently-deceased previous chief intermediary who had a genius for explaining the two sides to each other. The King is described as having "a personality like atonal music". He seems to want to be fairly benevolent but keeps messing up because he is bewildered by how humans think. For instance as part of his war with the King of Brazil he orders that "the contents of art galleries and museums be burned to make way for weapons shops" (I think "shops" means "workshops" here.) Yet the King is no fanatic; he can be talked out of these schemes quite easily. In that example it seems that he just did not understand why humans cared about works of art.
There follows quite a complicated plot involving a human rebellion timed to occur when the British aliens are having a naval battle with their Brazilian counterparts. The leader of the rebellion is a colleague of Smith's in the King's administration called Hotch. Hotch considers himself to be a human and British patriot and is a very dynamic character in contrast to Smith who has taken on some of the King's apathetic melancholy.
The rebellion succeeds. Hotch, ruthlessly and in violation of the alien-imposed laws of war, launches nuclear rockets at the cities of Brazil, presumably in an effort to eradicate the aliens ruling there rather than out of any particular enmity to the human population. The Brazilians launch nuclear-armed bombers to strike Britain in retaliation.
Hotch having clearly won, the King gives up. Because he feels that in some strange way he is British, he does all he can to help Hotch reactivate the old air defence system against the Brazilian bombers. But we never find out whether these efforts succeed. Smith, the narrator, has ceased to really care. The reason for his indifference is that the King has shown him space and in the face of its vastness...
what gets knocked into you is this: nothing matters. Nothing is big enough to matter. It is as simple as that.
In the last paragraph Smith says,
Like the King, I was quickly becoming incapable of judgement. But before it goes altogether, I will say this of you, Hotch: It was a low trick you played on the King. A low, dirty trick to play on a good man.
"Enter a Pilgrim" was the first story in Dickson's Shane Evert series, a series of four stories which was turned into the 1987 fix-up novel Way of the Pilgrim. The third story in the series, "House of Weapons", is available at the Internet Archive.
Here is Charles Ashbacher's Amazon customer review of Dickson's novel Way of the Pilgrim:
This is one of the best science fiction stories ever written. Several years before the tale begins, an alien race called the Aalaag arrived on the Earth and easily take control. Their technology was so far advanced over humans that all military resistance was futile, the most advanced human technology could not even reach the level of scratching their paint. The Aalaag are also a species with a strict code of honor, and their goal is to harness the resources of Earth to build the Aalaag strength so that they can eventually reclaim their worlds. Many centuries before, an even stronger species had taken over the Aalaag home systems, forcing them to flee out across space, looking for new places to live.
While the presence of the Aalaag has brought an end to war between humans and created a very ordered society, the Aalaag mentality is such that the humans are considered to be the equivalent of cattle. The main character, a linguist named Shane Evert, is one of the few humans capable of speaking and understanding the Aalaag language. He is a translator for the Aalaag governor of Earth, in some ways one of the highest ranking humans on Earth.
As the story begins, he witnesses an Earthman being killed by the Aalaag for an act they consider rebellion. An Aalaag youth unintentionally injured the man's wife so he attacked the Aalaag with his bare hands. According to Aalaag law, the man must immediately be put to death by being impaled on spikes and all humans in the area forced to watch until the man is clearly dead. Evert is repulsed and draws an image of a cloaked man with a staff under the dead man. With this act, he takes the first step in becoming the pilgrim, the worldwide symbol of human resistance to the occupation.
However, he knows that any overt resistance against the Aalaag is futile, so he must find a way to fight back without overtly challenging the Aalaag. As his plan develops, he creates a worldwide network of resistors, which grows to include the covert security services of the major nations. They all cooperate to prepare for the day when humans finally challenge the power of the Aalaag.
Shane uses his knowledge of the Aalaag to convince the governor that they will achieve no real value if they continue their hold over Earth. He is genuinely surprised when the Aalaag governor agrees and they abandon Earth without destroying any structures or killing any humans.
What makes this story so powerful is the interaction between the alien race and the humans. Even though the Aalaag governor and Shane talk at length about their differences and their similarities, and do find some common ground, in the end the governor still considers the humans to be ungovernable cattle. Dickson is superb in creating an ending that gives you pause. Instead of a joyous triumph at the human "victory" over such a powerful foe, it is very bittersweet. Human national rivalries resurface even before the Aalaag are gone and you think deeply when Evert is told that the reason the Aalaag are leaving is because they consider the human species to be unworthy. Despite their actions of enslaving the human race, the Aalaag are very honorable beings, and they have many admirable qualities.