If this works, eventually there won't be any more mosquitoes.
Which would be nice. There isn't an animal on Earth that relies on mosquitoes as their sole food source, and basically they exist just to cause misery.
That is the problem. It not only won't work, it cannot possibly work.
Eliminating mosquitoes in a specific area, regardless of the method used, is only a temporary solution. It only effects one generation of mosquitoes. Unless the same eradication method is used every year, it can never work. Even then it would be ineffective on the fringes, as populations of mosquitoes they haven't eliminated begin moving into the area that has been eliminated.
The only way to completely eradicate all mosquitoes is to sterilize them all - every last one of them - all at the same time, everywhere on the planet. Which obviously cannot be done.
It is also not true that animals do not eat mosquitoes. Many animals consume mosquitoes including birds, fish, bats, amphibians, reptiles, and of course other insects.
Furthermore, mosquito larvae eat algae, parasites, fungus, and other microorganisms, and they eat all the time. That means mosquito larvae can get through a lot of detritus (or biological waste) in just a few days. Mosquito larvae produce frass (insect crap) which plants find very nutritious.
Mosquitoes, like bees, are also pollinators. Their primary source of food is the nectar from flowers. In the Arctic especially, due to the lack of diversity in insect life (there are no bees that far north), mosquitoes play a significant role in creating diversity among the tundra north of the Arctic Circle.
