Tbh though, a lot of black people love chicken and have big lips.
A lot of white people are elitist or racist or both and pretend not to be, to the point where they even believe their own lies.
A lot of indian people smell of curry, because herp derp, a lot of indian people eat curry. Speaking of which, us westerners apparently stink of milk to the japanese. But milk doesn't smell! I don't notice it! Same for indians I guess lol.
A lot of slavic people drink like fish.
A lot of japanese people are xenophobic.
A lot of chinese people are good at mathematics.
I said all those things and I believe them because as much as you may pretend it's somehow offensive to say them, they are all true pretty much.
"A lot of ____ people ____" depending on the context is different from "_____ people like ____." Or, considering the original statement this blog is centered on: "I saw a ____ person coming out of Target. All he had was a ______." There is no punchline to that joke except the cliche.
Also, this discussion is different from an American perspective than a British one. Our own history of racial violence means that certain words and ideas will trigger reactions specific to us. Think of it like German society's views on Nazism. Slavery is America's own Nazi shadow, something that hangs on the collective consciousness.
Also, consider some of the observations/generalizations you mentioned and where they come from. "A lot of Chinese people are good at mathematics," for one. China is a big country, and it isn't necessarily true that Chinese people are more disproportionally good at math than people in any other industrializing society. What is more likely to be true is that the government controlled academic curriculum emphasizes math more than other subjects, and the students who get selected to travel outside of the country have to be good at math in order to study overseas. These Chinese students are the Chinese people that foreigners are probably most used to seeing and interacting with, which gives foreigners the impression that a lot of Chinese are good at math. But that observation is dependent on where you are in the world. If you were in China, you might not say that Chinese people are good at math.
The stereotype of the Ugly American is another example. Americans are characterized outside of their country as rich and lazy, and that makes sense if you consider that it takes money to travel, so only Americans who can afford to travel can be tourists in other countries. So the world isn't going to see a lot of poor or working-class Americans.