Which policy? The latest H1-B bill which lifts per-country caps passed unanimously in the senate and with a strong majority in the house.
https://www.congress.gov/bill/116th-congress/house-bill/1044
Certainly different people will have different opinions on what the "right" number of immigrants is but this is not an anti-immigrant country.
That's because it's designed to let in more rich Asians and less poor Latin Americans and Africans.
In terms of popular political support I would say that the United States is very anti-immigrant, and has been for a very long time. The reason the U.S. continues to let in so many immigrants is because of business interests, which have always held considerable political power. And the reason the U.S. has such diversity of immigrants is because of the peculiarity of our laws (e.g. by promoting so-called chain migration, a term Trump has turned into a pejorative), which have largely stayed the same because of inertia and political deadlock, but as you pointed out this is likely to change soon.
Also, the U.S. is no longer unique in terms of its large foreign born population. The U.K., for example, has roughly the same percentage. Brexit is no coincidence. Ditto for Germany. Though AFAIU German anti-immigration sentiment is mostly focused on immigrants from outside the E.U. (e.g. Turkey), whereas a large part of the sentiment in the U.K. is focused on Eastern European immigrants. Thus Angela Merkel has managed to placate voters by tacitly supporting harsher restrictions at the E.U. border, the short-lived Syrian influx notwithstanding. Brexit was driven in part by, e.g., people being pissed at low-wage Polish migrants.
I'll agree that the U.S. isn't worse than other countries. And I suppose our history as being a melting pot relatively unique unto the world is still worth something. (Though many parts of Latin America also saw similar migration flows at the turn of the 18th century. There are still German speaking towns in Brazil, Welsh speaking towns in Argentina, and a curious number of ethnic Japanese all over South America.) But we don't really stand out anymore one way or the other.
Immigration cap? Without a cap on H1B the US would have 10 or 50 times the level of incoming immigrants.
Plus H1B is limited to having degrees (or a lot of work experience) in a small subset of occupations.
The US could bring over 1 million doctors in one year if it were legal.
H1B is only one specific case of immigration: generally speaking the developed world allows for unconstrained immigration if you have a job offer. Americans are particularly restricted in hiring foreigners, which is also one reason why it has so much illegal immigration.
