You'll notice that Wikipedia article only discusses the Lump of Labor Fallacy in terms of Europe.
In the United States, things are a little different than the eurozone. You don't just get handed citizenship, you have to wait for years. Around 1/3rd of our immigrants are illegal, unskilled, and uneducated bumpkins with no meaningful English proficiency. Those folks have no chance of obtaining a loan, business license, or necessary permits, ever. And when you're not a citizen, you're paid in dirt and peanuts...and then the vast majority of legal immigrants will have no substantial capital by the time they gain citizenship, and any they do would go to immigration lawyers.
While some immigrants are absolutely job creators (I work for a company started by a naturalized citizen), the ratio of workers to job creators just isn't what you'd expect it to be.
Historically, it used to be the immigrants that created more jobs, but that was before the 1965 Immigration Act which increased abolished country of origin specific quotas. Most influx was restricted to 1st world western and northern european immigrants, who brought abundant capital, skill, and entrepreneurial spirit into the country. After the act, most of the immigration came from capital-poor and family-oriented (as opposed to entrepreneurial-oriented) folks from Asia, Africa, and south of the border.
The effect was quite immediate. Median wages adjusted for inflation started dropping in 1968 and have been trending downward ever since.
If we wan't to go back to a 1968-style economy and income distribution we're going to have to repeal the 1965 Immigration Act, and that's all there is to it.