Some may hope that if you do population control long enough, they eventually go extinct, but I think the evidence for that is pretty low.

We already eliminated Malaria carrying Mosquitos from large parts of the West with DDT and related techniques. Those mosquitos didn't manage to easily recolonize the areas from which they were driven away.

Louie Helm article suggest that SIT is enough to drive mosquito species to extinction. Do you think there a reason he's wrong? His numbers might be on the low end but spending a few billions would very much be worth it to eliminate all human biting mosquitos.