Lately I've read some interesting arguments in favour of highly stratified and inequitable societies, provided that mobility is at the very least a prospect or possibility, in fact even if it isnt but is something of a popular fantasy encouraged by things as few and far between as a lottery.
I would modify "America doesn't have class" to something more like "America does have class, but it also has class mobility."
...just not as much as one may think.
We've clearly increased our general level of affluence in the US since World War II. However, social class isn't about money, it's about opportunities and outcomes. The higher your social class, the greater autonomy you have from the power of others, and the greater influence your power has over others. What we call the "middle class" in the United States is actually a highly-affluent working class, relative to its forbears.