And since we can prove that immigration leads to less children being born, that means that future native born people who would otherwise exist are vanishing and being replaced by immigrants from other countries.

When that immigrant leaves their country, they will be replaced by the shockingly high birth rates. If Pakistan for example has one less mouth to feed from the slums, a new one can replace them.

This whole system is not too dissimilar to if we banned white parents from breeding and ordered that they adopt children from India instead. The difference is, they can't refuse, because they will be put in jail if they don't pay tax (goes towards these immigrants) and they can't breed because they'll suffer with no economic stability.
How does it feel?