Actually, the consensus among economists is that immigration makes your own economy better off. It has a net positive effect on productivity and prosperity.
In other words, your "rate-limiting" analogy makes zero sense. You want more immigration, not less.
just because the overall economy is better off (often measured in GDP) doesn't mean it is better off for all existing residents (e.g. GDP/capita).. even from a pure free-market perspective, distribution of wealth is not so market driven especially in the short term and when massive distortions deliberately exist in the marketplace (regulation, patents, near-monopolies, barriers, etc.)
Testing the limits of the analogy, but there you are.