Generally those efforts are targeted at disease carrying mosquitoes. There are other species of mosquitoes in the same regions and there aren't any animals that only eat one kind of mosquito.
These mosquitoes do some pollination but given that they're invasive it probably wouldn't be a big deal to wipe them out.
Also, invasiveness is true of a lot of pests in North America: the American cockroach isn't American, it's from Africa. Brown rats, red/fire ants, and wild pigs are also imported/invasive.