...is an argument against having any kind of temporary work visa at all...

Fine, I can jive with that. It's Congress' job to make those rules. Don't blame the people who are following the rules.

large numbers of immigrants extremely plausibly suppresses the native birth rate

Is this fact? Can you cite a source? This seems entirely made up. The idea that someone based their decision to have kids or not based on how many immigrants exist seems really far-fetched.