These are all excellent points. The increase in labor productivity accruing to immigrants to e.g. the US is often discussed by economists. I'll grant that it's not often discussed in general media, which is part of PhilGoetz's point, but I'm sure I've seen it there too.
Also, many economists have argued that in certain contexts immigration (even low-skilled) does result in economic gains for the native born. The argument goes that immigrants' negative impact on native born wages is small and that this small change is more than offset by the immigrants' ability to make domestic goods cheaper. People earn less, but things cost less still. In this scenario GDP per person has gone down, but native born purchasing power has increased. And the immigrants are far better off- their labor productivity has increased through the place premium and (related) their wages are almost certainly higher than they would be in their country of origin.
A final point:is that total GDP in some contexts can actually be a good in itself. Having a country with a large GDP is very meaningful politically. The fact the Norway and Qatar have higher GDP per capita than the US is meaningful and worthy of discussion, but the US matters far more to global politics and it's due primarily to one reason- the US is responsible for one quarter of the entire world's nominal GDP. Along some margin, allowing immigration that lowers GDP per capita but raises total GDP can be beneficial to members of a country purely based on international economic strength.
Related to your second paragraph, I was intrigued when I saw economists pointing out that even low-skilled immigrants could raise natives' productivity & income by (1) nudging natives to upskill and move into higher-skill jobs, and (2) lowering childcare & housework costs, making it easier for native women to work paid jobs.
I just saw another comment implying that immigration was good because it increased GDP. Over the years, I've seen many similar comments in the LW / transhumanist / etc bubble claiming that increasing a country's population is good because it increases its GDP. These are generally used in support of increasing either immigration or population growth.
It doesn't, however, make sense. People have attached a positive valence to certain words, then moved those words into new contexts. They did not figure out what they want to optimize and do the math.
I presume they want to optimize wealth or productivity per person. You wouldn't try to make Finland richer by absorbing China. Its GDP would go up, but its GDP per person would go way down.
I know why LWers want to say that increasing GDP is good. Historically, LessWrong is associated with transhumanism and specifically extropianism, and one of the main opponents of transhumanism is the romantic, anti-technology part of the environmentalist movement. The Extropians were strong supporters of Julian Simon, an economist who argued against environmentalists that population growth would not lead to collapse, but to lower rather than higher prices.
Growth is good. That was one of our mantras in the 1990s. And economists measure growth via GDP. So increasing GDP is good, right?
Wrong. Increasing GDP is not the same as productivity growth. Productivity makes sense either per person (when speaking locally) or per planet (or other isolated system, when looking at the big picture or the rate of technological change). GDP stands for Gross Domestic Product. It refers to the total yearly market value of the product of those people contained within some rather arbitrarily drawn border on a map, and is used for national budget planning. Both are surprisingly complex to compute, but in very different ways.
If you have a constant population, and GDP increases, productivity per person has increased. But if you have a border on a map enclosing some people, and you move it so it encloses more people, productivity hasn't increased. If instead of moving the border over the people, you move the people over the border, productivity still hasn't increased. Those people might be more productive on the other side of the border, but I haven't seen people make that argument. They just say increasing population increases GDP, so it's good.
If your population grows, your GDP will grow. That's not what Julian Simon was talking about. Simon argued that as population density rises, wealth per person will grow. He didn't say that increased GDP is a good thing, but that increased productivity is good. GDP is only correlated with productivity. (In fact, he argued against using even GDP per person as the sole measurement of growth: "Every time a human baby is born, the per capita GDP falls." Increased productivity makes prices fall, which makes wealth go up but GDP go down.)
Remember that the concepts of GDP and productivity are separate. You might be tempted to call GDP "productivity per country", but it just doesn't mean that when it's used in discussions about changing the population size or density of the country. It is invalid in that context to infer changes in productivity, and hence positive or negative valence, from changes in GDP. People would realize that if they remembered why they think increasing GDP is good, but I fear they don't--they just note the positive valence they've attached to the word and assume it's still valid in the current context.