Countless species are already going extinct all the time, both known to us and completely unknown to us, and both "naturally" and from human impact, and no one really cares.

Here's a rare case of a species that there is very good reason to deliberately get rid of, and suddenly people care and object, a lot!?

Of course there's justifiable skepticism and a long, sorry history of unintended consequences of introducing, eradicating or in any way modifying a species or its behavior, but seriously, read up on it, this is not one of those species, pretty much everyone who knows anything about it (including otherwise very environmentally minded people who as a rule would never, ever agree with deliberately eradicating anything) agrees it can be eradicated with no meaningful impact on the ecosystem. (and yes, that includes even the armchair criticism from cynical misanthropes who consider human deaths resulting from this species a good thing "because humans are bad for the environment" - while these mosquitoes do cause a lot of human deaths, the environmental "benefit" of those deaths are completely meaningless in the grand scheme of things)