Well, no, birth rates are low because the reproduction of labor is unpaid labor. Yes, development is associated with lower birthrates, but only because no developed country has ever seriously tried to make reproductive labor a real job. Doing so would decrease the size of the workforce for production of commodities.

Now you're totally right that the people migrating from the Global South are fleeing underdevelopment from imperialism, and that this is itself a factor of underdevelopment. What you haven't considered is why the imperial core limits migration.

Racism is part of it, but only part of the larger structural base. If they allowed unlimited migration the imperial core would be filled with people from the periphery as they flee underdevelopment. This would at once reduce the availability of labor in the periphery and raise the contradictions of imperialism by making peripheral concerns into domestic concerns.

Migrants influence the society they're part of, causing agitation against imperialism. This would, ultimately, destabilize the core and allow for development to resume without imperial meddling.