I'm puzzled by this. During the late 19th century a huge influx of European immigrants came to the USA seeking better lives. But the late 19th century was a period of rampant inequality in the United States. During the gilded age when all those immigrants were moving out of Europe, was America actually more socially mobile and economically better than Europe at the time?
So a major factor in immigration was the Great Depression of 1873 caused by an influx of American and Russian grain that lowered the price of grain and caused one of the most surefire capital investments - wheat futures - to begin to fail. Why is that important for European immigration? Because if the price of grain goes down, in most agricultural communities, wages decrease. So the situation resulted in extremely hard times for many in the more agricultural economies of Europe most of which were from Eastern Europe and Southern Europe.
Many of those coming to the United States could more reliably get wages working new industrial jobs than the more taciturn world of agriculture. As you say, the Gilded Age is a time of increased wealth disparity, but even still the cash economy in the US was much more stable than in areas like Southern Italy. One consequence of these declining economic circumstances was a rising crisis in food security.
To offer an example, Giovanni Vecchi found that working class southern Italians were frequently at a caloric deficit. Vecchi is a professor of economics at the University of Rome and his book Measuring Wellbeing: A History of Italian Living Standards seeks to use economic models and statistics in interesting ways to map the growth of development of Italy’s economy and measure its effect on what he calls the ‘wellbeing’ of Italians. His conclusion was that malnutrition was frequent and diseases related to malnutrition (palegra, scurvy) were not uncommon.
In many cases the particulars were even worse.
One example has two sources that Vecchi is working with. Firstly, the age, height, body weight, and lifestyle of eight individuals, each from the most common occupations in Naples collected by the Neapolitan Luigi Manfredi. Second, estimated caloric intake of various occupations in Naples against the caloric requirement to do such work. Errand boys, Servants, and Lazzorones (a kind of Neapolitan beggar) ran up to an average 12% daily caloric deficit. Masons and peddlers ran a deficit up to 29%.
I bring all this in to highlight that you are correct to note the grueling conditions and dim chances for social mobility in the US on a class basis, but circumstances for many immigrants were quite dire and migration was just as often about survival as social mobility.