First, I'd like to point that those are gender, not species. Those together add to thousands of species, and you forgot at least about Lutzomyia.
That said, I completely agree that we should get rid of them. And think it's more likely than you stated, if we can extinguish so many species, it can't be so much harder to do it on purpose. Currently we are barely trying, with techniques from the earlier XX century.
Well, we could extinguish the mosquitoes if we nuked and irradiated the world so thoroughly that we experienced a 10-year nuclear winter that would cause mass extinction. No more animals to feed on, no more breeding zones, hence no more mosquitoes!
But that would come with some undesirable side effects such as the eradication of our own species (except for a few living in hardened, self-sufficient underground shelters as in the "Silo" novel).
Arguably, there are too many humans, so perhaps it would make sense to reboot the world with a tiny core of eugenically superior men and women, plus seed stocks and animals to repopulate, and of course maintain massive DNA banks for future species re-creation and replenishment as needed.
Or, preferably, terraform Mars or Venus and selectively introduce only the flora and fauna that humans like, and hope some jerk doesn't accidentally or deliberately release ticks, mosquitoes, and other pests into this erstwhile perfect new Eden.
That said, I completely agree that we should get rid of them. And think it's more likely than you stated, if we can extinguish so many species, it can't be so much harder to do it on purpose. Currently we are barely trying, with techniques from the earlier XX century.