Yes, Denmark has nice health coverage and social security, but its not at all “The American Dream”.
It’s more like a safety net from the American Nightmare, but just like everywhere else, you need to be born with capital to be successful in a capitalist society.
The free education does give people a better chance to make it from rags to riches through studying and working, but it’ll still take about one generation to make it further than top middle-class. To really really make it, you’ll need money to invest in your own business. That part might be harder, because it’s a really dense market, not just for selling but also for employing other people etc.
Part of the American Dream actually requires other people to be worse off. Everyone can “work hard” if given the option. The difficulty in USA is that not everyone has the option. The difficulty in Denmark is that everyone has the option, so the competition is harder, so it requires even harder work luck.