While there are many measures of greatness, let us take one of the simplest, income. America does have one of the highest median household incomes in the world. It may not look as good if you consider income distribution, hours worked, benefits provided, or other security or quality of life issues, but it would still be very high. So why do Americans have such high incomes?
Culture and institutions undoubtedly have an effect. Immigration was proposed as an answer, but immigration is really an effect rather than a cause. People want to move to where they can earn higher incomes. One of the reasons they have higher incomes here is because there are fewer people for the available productive resources, natural and otherwise, relative to those elsewhere in the world. Immigration can be beneficial in bringing in resources, but only if it brings in more than it consumes or competes with others already here. Immigration is beneficial to immigrants or they would not chose to come, but is only beneficial to existing inhabitants if the immigrants are above the average inhabitant, whether in achievement, skills, or resources. Unlimited immigration would be a negative as population would rise to equalize endowments and incomes would fall towards average, until no one more would want to immigrate here. For some, all that matters is whether the immigrant is better off, but most will consider whether they are better off as well, both individually and collectively. Some, especially those above average, may be better off as there will be more below them, but they may not be if social cohesion disintegrates, neighborhoods deteriorate, and the country declines to third world status. What even will the immigrant have gained if they no longer have anywhere they would want to immigrate to or would not want to do so again? Immigration can be good, but there can be too much a good thing. The downside can be in the stagnation and lack of progress of country left behind.