So what specifically about D&D's presentation of orcs, compared to how they're presented in LotR, do you find to be distinctly American?
So keep in mind I didn't say Orcs are distinctly American, but "race."
This includes the way elves, halflings, humans, etc are portrayed.
In my own understanding of race, the whole idea of race really became Americanized at the beginning of the slave trade. Those in power in the Americas had to figure out who it was okay to enslave, and who it was not. They needed a legal definition, so they decided to use something called "race" (rather than wealth, or gender, or other factors that had been used to justify owning other humans in the past).
Legally now, people of one "race" could own people of another "race," but not the other way around.
In order to morally justify this, each race was given characteristics. People of X race were such-and-such, which is why they could own other human beings. People of Y race were such-and-such, which is why they deserved to be enslaved. Oh, and let's go ahead and say People of Z race are all this, and people of that race are all like that, as long as it continues the idea that Y race should be in power.
Even after slavery was abolished, this fictitious idea of race influenced laws regarding segregation, lynchings, housing, justice, voting... Every aspect of American culture.
We now know, scientifically, that this is all bull-puckey. It was all made up to justify slavery and keep people who defined themselves as "white" in power.
So where does this show up in D&D?
Humans are the "default race," and all the other races are defined by how they vary from humans. Sometimes this is physical (pointy ears, wings, etc).
Often though, other "races" vary from humans culturally. Dwarves are traditional, halflings are plucky, orcs are savage.
And if you are half of one of those races? You still carry those stereotypes. You are defined as a half-orc, a half-elf.
This, then, is what I find very American about race in D&D. It's used as an othering tool. It's used to conflate physical differences and cultural differences. And it's used to define how player characters get to treat others. And it's tied inexorably to skin color!
You can kill a drow because they are evil. How do you know they are evil? They are cursed with black skin!
Of course, this is not universal. But it is systemic. The very existence of Drizzt shows that in order for a Drow to be anything but evil, they have to turn against their whole society.
To me, race in D&D is very Americanized because it echoes so many ways race was used to enslave, kill, discriminate against, and segregate people based only on their ancestry or skin color. Race in D&D is Americanized because it conflates "dark" with "evil." And Race in D&D is Americanized because it's obsessed with bloodlines and ancestry.