That saying is more true theoretically than practically, these days. The chief caveat I would add is about the word "poor." The gap between rich people and poor people is widening in America, and the American Dream of home ownership is becoming unobtainable for even middle-class families. In San Francisco, for instance, the Department of Housing and Urban Development defined "low-income" in 2018 as an individual making $82,200 or less. For the rest of the US, this threshold was $12,140.
Though the original poem about tired, poor, and huddled masses referred to immigrants, it could be applied as aptly today to "tired" workers and "huddled masses" of US residents. The traditional path to the American Dream—saving, investing, and borrowing—does not seem open to many of them. They can't save enough money to invest, because they have lost their jobs, are underemployed, or are young workers hampered by student-loan debt. Healthcare costs keep rising, interest rates are going up, and all these changes can seem overwhelming to those living paycheck to paycheck. These people are working hard, sometimes doing two or three gigs or part-time jobs at once, but still struggle to make ends meet.
The American Dream costs money, unless you're very lucky. Most Americans have $10,000 or less in their savings accounts. They're busy sweating about how to survive retirement—if they can ever afford to retire—rather than thinking about attaining a bigger house, a fancier car, or a more luxurious lifestyle.
Those who do fulfill the American Dream today seem to hit big on something—a startup company or a lucky investment, for example—instead of climbing the ladder. However, each year you read just enough about people who have done so (climbed the ladder to socioeconomic success) that, like the lottery, you have enough hope to keep playing the game.