Well, the continental U.S. and certainly the budding metropolis of New York was a very different place than Nevis in the Caribbean. Then again, “America” as a country didn’t exist at the time.
The use of the word “immigrant” may be stretching the truth a little bit (but just a little bit) in the service of brevity, a good story, and yes, a political point.
But the point stands that America was a nation full of and built by immigrants, especially back then.
But worth noting too that even back then, families who had been established for longer (certainly as landowners or slaveholders) had a sort of privilege.