Hi again. I'm speaking only from the perspective of New York City here, that's all I know about.
NYC was a major port of entry for the United States in the 19th Century. You can read the history of immigration to this country in the birthplaces of the domestic servants in City households in the state and federal censuses. First they're Black, then German, then Irish, then Italian.
The Europeans came to the United States to escape oppression, religious fiscrimination, forced conscription in their dukes' wars and because they were starving. The Irish potato famine was a very real event beginning in the 1840's. There is a lovely memorial in the City down by the World Trade Center called the Irish Hunger Memorial. The numbers who died of starvation, neglect and disease in Ireland are staggering.
Irish immigration, for economic reasons, continued on into the 20th Century. It was a matter of necessity. Because it was easier for women to get employment as domestics than for men to find work as laborers, young women often came first and since they lived in, they sent a great deal of their money home until their husband or betrothed could come to the US. They all came with nothing and so had to find work. Because there always seemed to be wars in Europe (and rebellion in Ireland) and because there was forced conscription for the Germans at least, I don't think any immigrant was surprised at the nature of War over here.
Despite the hardship experienced by Union footsoldiers, no one made them go, they chose it and I think often with pride.
As I mentioned before, during the Civil War, surely a big part of the appeal to young single men arriving with but a dollar or so in hand was the bounty, the pay and the food and lodging. While outside of the city, men who knew each other served together because regiments were raised from neighboring counties, in the cities of Boston, New York and Philadelphia, men of the same ethnicity joined and served together, providing a group with similar backgrounds to share the experience as comrades.
As referenced by others, the Army grouped Irish regiments from NYC, Boston and Philadelphia together, into the Irish Brigade, in part because the Irish were often disdained as a group and non-Irish units didn't want to fight with them. To me, this doesn't add up to exploitation. As these young men were volunteering, young men who had been here longer were also volunteering in droves.
An interesting but difficult exercise would be to take the soldiers of the Irish Brigade and run the names through the ship's passenger lists arriving in NYC, Boston and Philadelphia, and see if it could be established in many cases how recently the soldiers had arrived in the US. What with many common names (e.g.James Ryan etc.), the fact that men didn't always enlist under their real names and the many missing or abbreviated passenger lists, the exercise could not be 100% successful. But maybe trying a sample would be worthwhile.