In many measures of quality of life some of the American colonies were significantly higher than in Europe. New England in particular is sometimes colloquially said to have “invented grandparents” because the people who resettled there on average added enough to their life expectancy that it became very common for most people to know their grandparents.
Many of the Hessians who were essentially forced to come fight in the United States (they were mercenaries in that George III paid their Prince money for their services, but they themselves were conscripts or enlisted men in their Army, they had no option to decline deployment), attempted to and many did stay to live here after the war because they found it far preferable to the conditions of Germany at the time.
As a rule, the United States until cross-Atlantic travel had declined tremendously in cost was settled somewhat by the middle class of Europe. The extremely rich in Europe, other than ones who had fallen into extreme political disfavor (like some Huguenots who emigrated to some of the Southern colonies), had little interest in or desire to move to colonial North America. The extremely poor had far less means of getting across the Atlantic. The middle class tended to thus make up a disproportionate share of the early settlement.
There were of course opportunities to move to the colonies if you were extremely poor–by entering an indenture contract, and that was a major source of labor in the early colonies, but nonetheless on average, most of settlement into Britain’s North American colonies from the 1600s, up through the Revolution and into the early 1800s, was made up of middle class people.
Around the 1830s and on prices started to decline quite a bit, and that increased in the 1860s and beyond as “steam and sail” ships became more common and started to drive trans-Atlantic costs down across the board (the very poorest would still be making the transit on sail ships, but the competition from steam clippers had a downward influence on crossing rates of all ships, if that makes sense.) As an example of how much prices had dropped even before steam and sail ships had started to dominate (although some were sailing at the time), in 1816 passage from Liverpool England to New York harbor was over 3 pounds. By 1846 it was only 30 shillings (reminder in pre-decimal British currency there were 20 shillings to a pound, so this was roughly a 50% nominal decrease, note that while much lower than modern eras there was still inflation in the 19th century so on top of the nominal decrease, it was a value for money increase of even more than 50%.)
Another thing that somewhat reinforced this is the British did not want people emigrating to the independent colonies. They actually maintained specific policies that generally kept trans-Atlantic travel cheaper if your destination was a British North America (Canadian) port versus an American one. The cheapest passage to North America in the early 19th century was actually booking extra space on a fishing vessel, these could take you to Nova Scotia or Newfoundland. While I don’t know the circumstances of their crossing, this is when my paternal line actually made it into North America–landing in Nova Scotia in the 1840s and from all genealogical records we can tell that branch of the family remained deeply poor as an ongoing state of affairs, the branch that I’m from crossed into the United States as farm laborers in the 1890s, and eventually ended up in the Appalachias owning small parcels of land.
So TLDR, early America had much less extreme poverty than Europe. How would this translate to something like a modern GINI or HDI rating, very hard to say.
On a number of other measures, America was historically a middling performer until the late 19th century. I don’t have GNP figures for the first half of the 19th century. But in 1830 as an example, from data compiled by Paul Kennedy in his famous “The Rise and Fall of the Great Powers”, there’s some interesting data on things like “Relative Share of World Manufacturing” and “Per-Capita Industrialization.”
In terms of “relative share of world manufacturing”, here are some comparison points:
United Kingdom - 4.3
France - 4.2
German States - 3.5
Italian States - 2.5
Russia - 5.6
United States - 0.8
China - 33.3
(Yes, as a share of manufacturing output even without even the beginnings of an industrial revolution, vast China was…much ahead of Europe in this metric.)
United Kingdom - 9.5
France - 5.2
German States - 3.5
Italian States - 2.3
Russia - 5.6
United States - 2.4
China - 29.8
You’re starting to see the UK pull away here, as it is starting to “win” the early Industrial Revolution.
I won’t go through every year breakdown, but by 1900 the United States was up to 23.6%–China was down to 6.2%, and Britain was a bit behind the United States at 18.5%. So in about 70 years the U.S. went from being 1/12th the share of global manufacturing output as China, to being 3.8x that of China. Note that China’s population still grew at a very high rate over these 70 years, but the high population growth rate without significant industrialization, very likely decreased quality of life and per capita wealth in China by the year 1900. There’s a reason conditions politically in China were “not good” by then.
Another interesting measure from that same book is “per capita industrialization”, with 100 on this metric = to the United Kingdom in the year 1900.
By that metric, in 1800:
United Kingdom - 16
France - 9
German States - 8
Italian States - 8
Russia - 6
United States - 9
China - 6
So while significantly behind the mother country, in 1800 the U.S. was already on a per-capita basis more industrialized than Russia, China, Italy, Germany and on par with France.
United Kingdom - 25
France - 12
German States - 9
Italian States - 8
Russia - 7
United States - 14
China - 6
By 1830 the U.S. was #2 in the world behind the UK. You might notice there’s also a correlation here with countries that don’t improve much in these ratings being countries that go through profound politilcal/economic/social changes. Germany for example (really the many small independent German polities) eventually unifies and massively industrializes, but until some of the reforms and improvements made possible by unification, it was not particularly impressive in this metric. Countries like China and Russia that were still operating large portions of their economy with agrarian peasantry, barely improved in this measure at all in 30 years time.
By 1900 the UK was at…100 as the measure would suggest, and the U.S. was #2 at 69. All other countries are significantly behind that–Germany the next highest at 52, then France at 39. China is 3 in 1900, demonstrating that its per-capita industrialization has declined again, due to very high population growth without and industrial revolution.
None of this actually probably answers the OP, but may be a decent foundation for exploration.