Huh.

If the population is really sparse, immigration really does improve the society -- it allows for specialization, with supermarkets, karate studios, and the like. Then, as the population gets more dense, the marginal benefit of population growth, assuming random average people, decreases. Eventually, it goes negative. At that point, you need to raise the bar -- e.g. only admit high quality immigrants, or family reunification (which benefits citizen family members). The higher the population, the higher the bar needs to get raised.

The problem with your analysis is that it doesn't take into account the total population number. Any advocacy for immigration that doesn't differentiate between a national population of 100 million, 350 million, and 900 million, if that advocacy argues that it benefits the country, is intrinsically defective,