For example look at the phenomena of keystone species. If you remove starfish from a beach, the mussels will take over and the rest of the tidepool animals get wiped out. The reason is that the starfish prefer eating mussels and so maintain a balance. Unless you really know the ecosystem, it is hard to tell what is a keystone species that completely changes things.
An introduced species is easier. Unless it is in the same ecological niche as a native, there is no possibility that it is a necessary part of the balance.
Unfortunately, mosquitoes are likely to have a significant ecological role. They have preferred targets. Their role as a disease vector affects what population density those targets can have. Wipe out the mosquitoes, and you don't know what will happen to that ecosystem.
That said, I've been bitten by enough mosquitoes that I'm willing to risk it...
The methods discussed in the article can be targeted on a per species basis. The action could be limited to mosquito species that prefer to target humans.
(the sterile male method is already being used)
But to explain a bit of the reasoning, as I understand it, invasive species are considered harmful to ecosystems because they tend to throw them out of equilibrium. If you introduce something to an environment with no natural predators or other environment feedback loops to constrain population growth, the newly introduced species has a field day, for a bit, until it kills off whatever it feeds on. Since ecosystems are inherently deeply interconnected things, the boom-bust cycle causes all sorts of problems.
Now maybe a new equilibrium is eventually "better" than it was before, but that, I think, is the real arbitrary human value judgment. Observing that introducing a new species throws existing systems out of whack isn't a value judgement - it accurately describes what happens.
If I'm understanding your comment about the rate of change, I think you're talking about drift between equilibria.
Asking about "too fast" is where the human value judgement comes in. Which isn't invalid, it is just human-centric and far from the whole picture. Stating that introducing an invasive species can destroy the existing ecosystem is a statement of fact without judgement. It is just an observation of reality.
Typically zoologists, botanists, and other biologists who study local ecosystems and determine the damage done by invasive species.
Construing this as a human concept doesn't make it arbitrary - anthropogenic impact on the environment is important because we rely on our environment for survival as a species. Damage isn't limited to just a less biological diversity, it can and has stretched into agriculture and human production.
And back on topic - Kill 'em all - native, invasive, makes no difference. As I read the expert concensus, after eradication the likelihood of some edge case eco-dependency being revealed should be vanishingly small, whereas the status quo allows millions of homo sapiens to be sickened and or killed.
That seems like easy math to me.
NB: I'm a Southerner (Peach state) so I'll admit I carry a strong, but well earned, hatred of the vile creatures.