We wouldn't have to destroy all mosquitos, though. Only a few species that are carriers of human pathogens. Once they are gone, other, benign mosquitos will occupy their spot, and will be eaten by the same animals as before.

Such a vague blanket statement sounds like an article of faith and not an actual informed opinion about the specific species of mosquitos, the animals that eat them, and the fungibility of their food sources. Correct me if I'm wrong though.