Americans aren't pretending because unlike England, the country itself was created without a monarchy handing out titles of nobility[1]. Europeans can trace deep historical bloodlines -- which further reinforces class distinctions and self-awareness across generations. However, America was built on immigrants arriving on ships where family records were lost.
So if one still insists on overlaying the word "classes" on top of America, it is very different from "class" in Europe. Therefore, social class in America is more about wealth rather than hereditary titles. The ability for money (e.g. become a millionaire/billionaire) to change your status in society means social class in the USA is fluid and can change.
Maybe a loose USA version of hereditary class might be families like Kennedys, Bushes, etc where the media nicknames the Kennedys "Camelot". Even so, a bricklayer who was the son of an Austrian village policeman and had a bloodline tainted by a bastard great-grandfather -- was able to marry into the Kennedy family. Why? Because Arnold Schwarzenegger became a successful multi-millionaire. The European-version of "lower class" in Arnold's family background wasn't a barrier in the USA.
>Everyone pretends that racism in America is just that, when it's actually classism born of racist roots
I think many do have pure and simple racism based on skin color. E.g. they don't care for black billionaires like Michael Jordan and they think it's wrong that a black judge sits on the Supreme Court. Yes, racism for some people may be (subconsciously) intertwined with class as you point out ... but for others, it's a simple dislike of black people.
[1] https://en.wikipedia.org/wiki/Peerages_in_the_United_Kingdom