This is just plain false. Them and their eggs are an important food source for all kinds of aquatic animals and birds. Also, their larva remove contamins for water.
Even logically this doesn't make sense. What's more likely: every insect is extremely important except one, or humans are wrong (as we so often are) about one species of insect?
Multiple people in opal mining areas of Australia and at least one dude in a ghost town near death valley and at least one hotel off the top of my head. I'm sure there are more
I wonder how much has been studied about the ability of the other species to proliferate enough to make up and not be eaten out of existence. The mosquitos existing may not be the limiting factor on the population size of other species.