Other countries tend to have less productive economies than the US. Both overall and per worker. Of the few with higher productivity, most of them have more generous immigration policy than the US has had in recent years.

So yes, you are wrong, both about the impact of immigration here and the impact of closed borders elsewhere.

With the training I see it as a utilitarian issue. If training someone is a net benefit to the US I don't care where they were born. I certainly don't want to leave the benefit sitting on the table just because they weren't born here.